{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Linear codes for SSS-based polynomial masking\n", "\n", "### The SSS-based masking is a special case of GCM (Generalized Code-based masking)\n", "### $n=5$ shares, $t=2$, $\\ell=4$ bits: (5,2)-SSS (Shamir's Secret Sharing)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Parameters:\n", "\n", "- $Z=(X + Y_1\\alpha_1+Y_2\\alpha_1^2, X + Y_1\\alpha_2+Y_2\\alpha_2^2, X + Y_1\\alpha_3+Y_2\\alpha_3^2, X + Y_1\\alpha_4+Y_2\\alpha_4^2, X + Y_1\\alpha_5+Y_2\\alpha_5^2,)=X\\mathbf{G} + Y\\mathbf{H}$ where $X, Y=(Y_1, Y_2)$ and $Z$ are the sensitive variable, a mask and the protected variable, respectively, where $\\alpha_i$ for $1\\leq i\\leq 3$ are three public points in SSS-scheme\n", "- $\\mathbf{G} = [1, 1, 1, 1, 1]$ and $\\mathbf{H} = \\left(\\begin{matrix} \\alpha_1 & \\alpha_2 & \\alpha_3 & \\alpha_4 & \\alpha_5 \\\\ \\alpha_1^2 & \\alpha_2^2 & \\alpha_3^2 & \\alpha_4^2 & \\alpha_5^2 \\end{matrix}\\right)$ are two generator matrices of codes $\\mathcal{C}$ and $\\mathcal{D}$, resp.\n", "- $\\alpha_i\\in \\mathbb{F}_{2^\\ell}\\backslash\\{0\\}$ and $\\alpha_1\\neq\\alpha_2\\neq\\alpha_3\\neq\\alpha_4\\neq\\alpha_5$, thus there are ${15}\\choose{5}$=3003 linear codes for (5,2)-SSS\n", "- Each nonzero element over $\\mathbb{F}_{2^\\ell}$ can be denoted as $\\alpha^i$ where $i\\in\\{0, 1, \\ldots, 14\\}$, the corresponding irreducible polynomial is $g(\\alpha) = \\alpha^4 + \\alpha +1$\n", "- Due to equivalence of linear codes, we simplify the enumeration by choosing $(\\alpha_1, \\alpha_2, \\alpha_3, \\alpha_4, \\alpha_5)=(\\alpha^i, \\alpha^j, \\alpha^k, \\alpha^l,\\alpha^r)$ where $i=0$ and $0**We only show the first 500 codes because of the limit on file size.**." ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "ExecuteTime": { "end_time": "2020-10-18T20:19:55.950647Z", "start_time": "2020-10-18T20:19:53.508533Z" }, "scrolled": false }, "outputs": [ 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"
Tab. I All linear codes for (5,2)-SSS-based masking with $n=5$ shares over $\\mathbb{F}_{2^4}$.
$\\alpha_2$ $\\alpha_3$ $\\alpha_4$ $\\alpha_5$ $d_{\\mathcal{D}}^\\perp$ $B_{d_{\\mathcal{D}}^\\perp}$ Weight Enumerators* (the first 24 elements)
0$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{4}$34[0, 1, 3, 4, 4, 14, 5, 63, 6, 156, 7, 302, 8, 500, 9, 645, 10, 712, 11, 680, 1, 2, 3, 5]
1$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{5}$34[0, 1, 3, 4, 4, 13, 5, 67, 6, 165, 7, 288, 8, 477, 9, 663, 10, 735, 11, 668, 1, 2, 3, 6]
2$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{6}$36[0, 1, 3, 6, 4, 14, 5, 61, 6, 157, 7, 288, 8, 502, 9, 683, 10, 695, 11, 638, 1, 2, 3, 7]
3$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{7}$33[0, 1, 3, 3, 4, 16, 5, 60, 6, 155, 7, 315, 8, 490, 9, 644, 10, 723, 11, 649, 1, 2, 3, 8]
4$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{8}$36[0, 1, 3, 6, 4, 12, 5, 61, 6, 166, 7, 288, 8, 486, 9, 683, 10, 710, 11, 638, 1, 2, 3, 9]
5$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{9}$34[0, 1, 3, 4, 4, 15, 5, 64, 6, 150, 7, 296, 8, 515, 9, 660, 10, 692, 11, 660, 1, 2, 3, 10]
6$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{10}$418[0, 1, 4, 18, 5, 72, 6, 145, 7, 300, 8, 510, 9, 644, 10, 703, 11, 664, 12, 506, 1, 2, 3, 11]
7$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{11}$31[0, 1, 3, 1, 4, 17, 5, 70, 6, 148, 7, 301, 8, 503, 9, 642, 10, 720, 11, 667, 1, 2, 3, 12]
8$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{12}$31[0, 1, 3, 1, 4, 17, 5, 67, 6, 152, 7, 305, 8, 494, 9, 651, 10, 720, 11, 651, 1, 2, 3, 13]
9$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{13}$421[0, 1, 4, 21, 5, 63, 6, 152, 7, 312, 8, 479, 9, 655, 10, 728, 11, 648, 12, 503, 1, 2, 3, 14]
10$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{14}$34[0, 1, 3, 4, 4, 14, 5, 63, 6, 156, 7, 302, 8, 500, 9, 645, 10, 712, 11, 680, 1, 2, 4, 5]
11$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{5}$38[0, 1, 3, 8, 4, 13, 5, 49, 6, 165, 7, 316, 8, 477, 9, 653, 10, 735, 11, 648, 1, 2, 4, 6]
12$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{6}$37[0, 1, 3, 7, 4, 14, 5, 52, 6, 155, 7, 311, 8, 504, 9, 660, 10, 707, 11, 645, 1, 2, 4, 7]
13$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{7}$36[0, 1, 3, 6, 4, 10, 5, 58, 6, 178, 7, 302, 8, 456, 9, 658, 10, 750, 11, 658, 1, 2, 4, 8]
14$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{8}$39[0, 1, 3, 9, 4, 12, 5, 46, 6, 166, 7, 313, 8, 480, 9, 670, 10, 732, 11, 635, 1, 2, 4, 9]
15$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{9}$34[0, 1, 3, 4, 4, 11, 5, 62, 6, 170, 7, 304, 8, 475, 9, 650, 10, 732, 11, 660, 1, 2, 4, 10]
16$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{10}$38[0, 1, 3, 8, 4, 13, 5, 52, 6, 162, 7, 300, 8, 485, 9, 680, 10, 732, 11, 640, 1, 2, 4, 11]
17$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{11}$35[0, 1, 3, 5, 4, 13, 5, 60, 6, 160, 7, 301, 8, 495, 9, 660, 10, 712, 11, 655, 1, 2, 4, 12]
18$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{12}$35[0, 1, 3, 5, 4, 14, 5, 58, 6, 160, 7, 303, 8, 488, 9, 672, 10, 720, 11, 627, 1, 2, 4, 13]
19$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{13}$35[0, 1, 3, 5, 4, 14, 5, 58, 6, 158, 7, 309, 8, 490, 9, 650, 10, 732, 11, 655, 1, 2, 4, 14]
20$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{14}$34[0, 1, 3, 4, 4, 13, 5, 67, 6, 165, 7, 288, 8, 477, 9, 663, 10, 735, 11, 668, 1, 2, 5, 6]
21$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{5}$$\\alpha^{6}$36[0, 1, 3, 6, 4, 13, 5, 59, 6, 165, 7, 298, 8, 477, 9, 663, 10, 735, 11, 658, 1, 2, 5, 7]
22$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{5}$$\\alpha^{7}$37[0, 1, 3, 7, 4, 14, 5, 56, 6, 162, 7, 299, 8, 478, 9, 668, 10, 740, 11, 653, 1, 2, 5, 8]
23$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{5}$$\\alpha^{8}$38[0, 1, 3, 8, 4, 12, 5, 53, 6, 172, 7, 304, 8, 464, 9, 661, 10, 738, 11, 656, 1, 2, 5, 9]
24$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{5}$$\\alpha^{9}$35[0, 1, 3, 5, 4, 16, 5, 62, 6, 154, 7, 297, 8, 488, 9, 658, 10, 740, 11, 663, 1, 2, 5, 10]
25$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{5}$$\\alpha^{10}$38[0, 1, 3, 8, 4, 11, 5, 50, 6, 170, 7, 308, 8, 475, 9, 670, 10, 732, 11, 640, 1, 2, 5, 11]
26$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{5}$$\\alpha^{11}$35[0, 1, 3, 5, 4, 15, 5, 61, 6, 159, 7, 301, 8, 477, 9, 653, 10, 755, 11, 663, 1, 2, 5, 12]
27$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{5}$$\\alpha^{12}$34[0, 1, 3, 4, 4, 14, 5, 60, 6, 164, 7, 308, 8, 468, 9, 652, 10, 760, 11, 652, 1, 2, 5, 13]
28$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{5}$$\\alpha^{13}$35[0, 1, 3, 5, 4, 15, 5, 56, 6, 155, 7, 311, 8, 497, 9, 662, 10, 715, 11, 627, 1, 2, 5, 14]
29$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{5}$$\\alpha^{14}$36[0, 1, 3, 6, 4, 14, 5, 61, 6, 157, 7, 288, 8, 502, 9, 683, 10, 695, 11, 638, 1, 2, 6, 7]
30$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{6}$$\\alpha^{7}$34[0, 1, 3, 4, 4, 11, 5, 64, 6, 164, 7, 296, 8, 500, 9, 660, 10, 696, 11, 660, 1, 2, 6, 8]
31$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{6}$$\\alpha^{8}$37[0, 1, 3, 7, 4, 14, 5, 55, 6, 162, 7, 305, 8, 478, 9, 653, 10, 740, 11, 673, 1, 2, 6, 9]
32$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{6}$$\\alpha^{9}$32[0, 1, 3, 2, 4, 17, 5, 67, 6, 154, 7, 296, 8, 481, 9, 669, 10, 748, 11, 634, 1, 2, 6, 10]
33$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{6}$$\\alpha^{10}$32[0, 1, 3, 2, 4, 17, 5, 66, 6, 150, 7, 302, 8, 501, 9, 654, 10, 708, 11, 654, 1, 2, 6, 11]
34$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{6}$$\\alpha^{11}$37[0, 1, 3, 7, 4, 15, 5, 54, 6, 151, 7, 301, 8, 509, 9, 680, 10, 707, 11, 625, 1, 2, 6, 12]
35$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{6}$$\\alpha^{12}$32[0, 1, 3, 2, 4, 22, 5, 53, 6, 147, 7, 330, 8, 488, 9, 645, 10, 723, 11, 638, 1, 2, 6, 13]
36$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{6}$$\\alpha^{13}$36[0, 1, 3, 6, 4, 12, 5, 58, 6, 164, 7, 298, 8, 490, 9, 670, 10, 712, 11, 650, 1, 2, 6, 14]
37$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{6}$$\\alpha^{14}$33[0, 1, 3, 3, 4, 16, 5, 60, 6, 155, 7, 315, 8, 490, 9, 644, 10, 723, 11, 649, 1, 2, 7, 8]
38$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{7}$$\\alpha^{8}$36[0, 1, 3, 6, 4, 10, 5, 59, 6, 173, 7, 292, 8, 474, 9, 685, 10, 727, 11, 630, 1, 2, 7, 9]
39$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{7}$$\\alpha^{9}$36[0, 1, 3, 6, 4, 12, 5, 57, 6, 166, 7, 302, 8, 480, 9, 665, 10, 732, 11, 650, 1, 2, 7, 10]
40$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{7}$$\\alpha^{10}$32[0, 1, 3, 2, 4, 14, 5, 69, 6, 161, 7, 292, 8, 482, 9, 667, 10, 735, 11, 642, 1, 2, 7, 11]
41$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{7}$$\\alpha^{11}$33[0, 1, 3, 3, 4, 17, 5, 59, 6, 155, 7, 315, 8, 483, 9, 651, 10, 731, 11, 641, 1, 2, 7, 12]
42$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{7}$$\\alpha^{12}$36[0, 1, 3, 6, 4, 13, 5, 56, 6, 160, 7, 308, 8, 495, 9, 650, 10, 712, 11, 670, 1, 2, 7, 13]
43$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{7}$$\\alpha^{13}$32[0, 1, 3, 2, 4, 16, 5, 63, 6, 154, 7, 314, 8, 496, 9, 639, 10, 708, 11, 654, 1, 2, 7, 14]
44$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{7}$$\\alpha^{14}$36[0, 1, 3, 6, 4, 12, 5, 61, 6, 166, 7, 288, 8, 486, 9, 683, 10, 710, 11, 638, 1, 2, 8, 9]
45$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{8}$$\\alpha^{9}$34[0, 1, 3, 4, 4, 15, 5, 67, 6, 154, 7, 288, 8, 501, 9, 663, 10, 710, 11, 668, 1, 2, 8, 10]
46$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{8}$$\\alpha^{10}$38[0, 1, 3, 8, 4, 11, 5, 50, 6, 170, 7, 308, 8, 475, 9, 670, 10, 732, 11, 640, 1, 2, 8, 11]
47$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{8}$$\\alpha^{11}$36[0, 1, 3, 6, 4, 17, 5, 54, 6, 152, 7, 316, 8, 491, 9, 640, 10, 728, 11, 670, 1, 2, 8, 12]
48$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{8}$$\\alpha^{12}$34[0, 1, 3, 4, 4, 12, 5, 61, 6, 163, 7, 308, 8, 494, 9, 645, 10, 707, 11, 660, 1, 2, 8, 13]
49$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{8}$$\\alpha^{13}$34[0, 1, 3, 4, 4, 11, 5, 62, 6, 167, 7, 304, 8, 489, 9, 650, 10, 707, 11, 660, 1, 2, 8, 14]
50$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{8}$$\\alpha^{14}$34[0, 1, 3, 4, 4, 15, 5, 64, 6, 150, 7, 296, 8, 515, 9, 660, 10, 692, 11, 660, 1, 2, 9, 10]
51$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{9}$$\\alpha^{10}$32[0, 1, 3, 2, 4, 17, 5, 70, 6, 148, 7, 288, 8, 503, 9, 672, 10, 720, 11, 642, 1, 2, 9, 11]
52$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{9}$$\\alpha^{11}$35[0, 1, 3, 5, 4, 14, 5, 62, 6, 159, 7, 297, 8, 492, 9, 658, 10, 715, 11, 663, 1, 2, 9, 12]
53$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{9}$$\\alpha^{12}$31[0, 1, 3, 1, 4, 22, 5, 57, 6, 162, 7, 303, 8, 454, 9, 699, 10, 740, 11, 615, 1, 2, 9, 13]
54$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{9}$$\\alpha^{13}$31[0, 1, 3, 1, 4, 15, 5, 71, 6, 156, 7, 293, 8, 496, 9, 659, 10, 712, 11, 659, 1, 2, 9, 14]
55$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{9}$$\\alpha^{14}$418[0, 1, 4, 18, 5, 72, 6, 145, 7, 300, 8, 510, 9, 644, 10, 703, 11, 664, 12, 506, 1, 2, 10, 11]
56$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{10}$$\\alpha^{11}$33[0, 1, 3, 3, 4, 15, 5, 68, 6, 149, 7, 291, 8, 519, 9, 660, 10, 687, 11, 665, 1, 2, 10, 12]
57$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{10}$$\\alpha^{12}$417[0, 1, 4, 17, 5, 70, 6, 151, 7, 308, 8, 495, 9, 634, 10, 723, 11, 664, 12, 491, 1, 2, 10, 13]
58$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{10}$$\\alpha^{13}$31[0, 1, 3, 1, 4, 17, 5, 71, 6, 153, 7, 291, 8, 485, 9, 669, 10, 743, 11, 639, 1, 2, 10, 14]
59$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{10}$$\\alpha^{14}$31[0, 1, 3, 1, 4, 17, 5, 70, 6, 148, 7, 301, 8, 503, 9, 642, 10, 720, 11, 667, 1, 2, 11, 12]
60$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{11}$$\\alpha^{12}$37[0, 1, 3, 7, 4, 15, 5, 52, 6, 152, 7, 311, 8, 505, 9, 660, 10, 712, 11, 645, 1, 2, 11, 13]
61$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{11}$$\\alpha^{13}$31[0, 1, 3, 1, 4, 20, 5, 64, 6, 140, 7, 317, 8, 512, 9, 636, 10, 706, 11, 651, 1, 2, 11, 14]
62$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{11}$$\\alpha^{14}$31[0, 1, 3, 1, 4, 17, 5, 67, 6, 152, 7, 305, 8, 494, 9, 651, 10, 720, 11, 651, 1, 2, 12, 13]
63$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{12}$$\\alpha^{13}$34[0, 1, 3, 4, 4, 11, 5, 62, 6, 167, 7, 304, 8, 489, 9, 650, 10, 707, 11, 660, 1, 2, 12, 14]
64$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{12}$$\\alpha^{14}$421[0, 1, 4, 21, 5, 63, 6, 152, 7, 312, 8, 479, 9, 655, 10, 728, 11, 648, 12, 503, 1, 2, 13, 14]
65$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{13}$$\\alpha^{14}$34[0, 1, 3, 4, 4, 14, 5, 63, 6, 156, 7, 302, 8, 500, 9, 645, 10, 712, 11, 680, 1, 3, 4, 5]
66$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{4}$$\\alpha^{5}$34[0, 1, 3, 4, 4, 11, 5, 62, 6, 167, 7, 304, 8, 489, 9, 650, 10, 707, 11, 660, 1, 3, 4, 6]
67$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{4}$$\\alpha^{6}$37[0, 1, 3, 7, 4, 14, 5, 55, 6, 157, 7, 303, 8, 502, 9, 663, 10, 695, 11, 653, 1, 3, 4, 7]
68$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{4}$$\\alpha^{7}$39[0, 1, 3, 9, 4, 15, 5, 47, 6, 154, 7, 309, 8, 495, 9, 675, 10, 732, 11, 635, 1, 3, 4, 8]
69$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{4}$$\\alpha^{8}$35[0, 1, 3, 5, 4, 10, 5, 58, 6, 174, 7, 309, 8, 470, 9, 650, 10, 732, 11, 655, 1, 3, 4, 9]
70$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{4}$$\\alpha^{9}$310[0, 1, 3, 10, 4, 15, 5, 40, 6, 154, 7, 326, 8, 495, 9, 660, 10, 732, 11, 630, 1, 3, 4, 10]
71$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{4}$$\\alpha^{10}$34[0, 1, 3, 4, 4, 12, 5, 59, 6, 167, 7, 312, 8, 482, 9, 647, 10, 715, 11, 652, 1, 3, 4, 11]
72$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{4}$$\\alpha^{11}$34[0, 1, 3, 4, 4, 11, 5, 63, 6, 166, 7, 300, 8, 493, 9, 655, 10, 702, 11, 660, 1, 3, 4, 12]
73$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{4}$$\\alpha^{12}$36[0, 1, 3, 6, 4, 14, 5, 53, 6, 156, 7, 320, 8, 500, 9, 635, 10, 712, 11, 670, 1, 3, 4, 13]
74$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{4}$$\\alpha^{13}$34[0, 1, 3, 4, 4, 16, 5, 61, 6, 153, 7, 310, 8, 500, 9, 635, 10, 703, 11, 680, 1, 3, 4, 14]
75$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{4}$$\\alpha^{14}$38[0, 1, 3, 8, 4, 13, 5, 49, 6, 165, 7, 316, 8, 477, 9, 653, 10, 735, 11, 648, 1, 3, 5, 6]
76$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{5}$$\\alpha^{6}$32[0, 1, 3, 2, 4, 17, 5, 67, 6, 148, 7, 302, 8, 503, 9, 647, 10, 720, 11, 662, 1, 3, 5, 7]
77$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{5}$$\\alpha^{7}$32[0, 1, 3, 2, 4, 17, 5, 64, 6, 152, 7, 312, 8, 491, 9, 634, 10, 728, 11, 674, 1, 3, 5, 8]
78$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{5}$$\\alpha^{8}$417[0, 1, 4, 17, 5, 68, 6, 164, 7, 294, 8, 471, 9, 678, 10, 720, 11, 636, 12, 515, 1, 3, 5, 9]
79$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{5}$$\\alpha^{9}$34[0, 1, 3, 4, 4, 14, 5, 61, 6, 153, 7, 310, 8, 514, 9, 635, 10, 687, 11, 680, 1, 3, 5, 10]
80$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{5}$$\\alpha^{10}$34[0, 1, 3, 4, 4, 11, 5, 63, 6, 170, 7, 298, 8, 475, 9, 665, 10, 732, 11, 640, 1, 3, 5, 11]
81$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{5}$$\\alpha^{11}$34[0, 1, 3, 4, 4, 13, 5, 62, 6, 158, 7, 304, 8, 505, 9, 650, 10, 692, 11, 660, 1, 3, 5, 12]
82$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{5}$$\\alpha^{12}$31[0, 1, 3, 1, 4, 16, 5, 69, 6, 152, 7, 305, 8, 498, 9, 637, 10, 720, 11, 667, 1, 3, 5, 13]
83$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{5}$$\\alpha^{13}$31[0, 1, 3, 1, 4, 13, 5, 71, 6, 168, 7, 293, 8, 466, 9, 659, 10, 752, 11, 659, 1, 3, 5, 14]
84$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{5}$$\\alpha^{14}$37[0, 1, 3, 7, 4, 14, 5, 52, 6, 155, 7, 311, 8, 504, 9, 660, 10, 707, 11, 645, 1, 3, 6, 7]
85$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{6}$$\\alpha^{7}$33[0, 1, 3, 3, 4, 15, 5, 65, 6, 154, 7, 299, 8, 503, 9, 657, 10, 700, 11, 657, 1, 3, 6, 8]
86$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{6}$$\\alpha^{8}$32[0, 1, 3, 2, 4, 18, 5, 57, 6, 165, 7, 312, 8, 458, 9, 675, 10, 743, 11, 618, 1, 3, 6, 9]
87$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{6}$$\\alpha^{9}$36[0, 1, 3, 6, 4, 14, 5, 55, 6, 158, 7, 310, 8, 490, 9, 655, 10, 732, 11, 650, 1, 3, 6, 10]
88$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{6}$$\\alpha^{10}$33[0, 1, 3, 3, 4, 13, 5, 65, 6, 160, 7, 299, 8, 498, 9, 657, 10, 704, 11, 657, 1, 3, 6, 11]
89$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{6}$$\\alpha^{11}$36[0, 1, 3, 6, 4, 14, 5, 57, 6, 156, 7, 300, 8, 500, 9, 675, 10, 712, 11, 630, 1, 3, 6, 12]
90$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{6}$$\\alpha^{12}$33[0, 1, 3, 3, 4, 19, 5, 58, 6, 145, 7, 319, 8, 503, 9, 646, 10, 711, 11, 641, 1, 3, 6, 13]
91$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{6}$$\\alpha^{13}$36[0, 1, 3, 6, 4, 15, 5, 58, 6, 158, 7, 302, 8, 483, 9, 658, 10, 740, 11, 658, 1, 3, 6, 14]
92$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{6}$$\\alpha^{14}$36[0, 1, 3, 6, 4, 10, 5, 58, 6, 178, 7, 302, 8, 456, 9, 658, 10, 750, 11, 658, 1, 3, 7, 8]
93$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{7}$$\\alpha^{8}$31[0, 1, 3, 1, 4, 18, 5, 65, 6, 150, 7, 319, 8, 486, 9, 619, 10, 748, 11, 679, 1, 3, 7, 9]
94$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{7}$$\\alpha^{9}$310[0, 1, 3, 10, 4, 16, 5, 45, 6, 154, 7, 310, 8, 488, 9, 673, 10, 740, 11, 638, 1, 3, 7, 10]
95$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{7}$$\\alpha^{10}$31[0, 1, 3, 1, 4, 20, 5, 62, 6, 152, 7, 311, 8, 486, 9, 648, 10, 720, 11, 671, 1, 3, 7, 11]
96$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{7}$$\\alpha^{11}$31[0, 1, 3, 1, 4, 17, 5, 62, 6, 161, 7, 309, 8, 485, 9, 658, 10, 695, 11, 651, 1, 3, 7, 12]
97$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{7}$$\\alpha^{12}$32[0, 1, 3, 2, 4, 16, 5, 63, 6, 155, 7, 314, 8, 490, 9, 639, 10, 723, 11, 654, 1, 3, 7, 13]
98$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{7}$$\\alpha^{13}$31[0, 1, 3, 1, 4, 19, 5, 63, 6, 149, 7, 323, 8, 483, 9, 621, 10, 751, 11, 671, 1, 3, 7, 14]
99$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{7}$$\\alpha^{14}$39[0, 1, 3, 9, 4, 12, 5, 46, 6, 166, 7, 313, 8, 480, 9, 670, 10, 732, 11, 635, 1, 3, 8, 9]
100$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{8}$$\\alpha^{9}$34[0, 1, 3, 4, 4, 16, 5, 64, 6, 154, 7, 296, 8, 494, 9, 660, 10, 718, 11, 660, 1, 3, 8, 10]
101$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{8}$$\\alpha^{10}$34[0, 1, 3, 4, 4, 10, 5, 63, 6, 172, 7, 298, 8, 480, 9, 665, 10, 712, 11, 640, 1, 3, 8, 11]
102$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{8}$$\\alpha^{11}$31[0, 1, 3, 1, 4, 13, 5, 71, 6, 168, 7, 293, 8, 466, 9, 659, 10, 752, 11, 659, 1, 3, 8, 12]
103$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{8}$$\\alpha^{12}$31[0, 1, 3, 1, 4, 19, 5, 64, 6, 148, 7, 317, 8, 489, 9, 636, 10, 736, 11, 651, 1, 3, 8, 13]
104$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{8}$$\\alpha^{13}$34[0, 1, 3, 4, 4, 11, 5, 62, 6, 167, 7, 304, 8, 489, 9, 650, 10, 707, 11, 660, 1, 3, 8, 14]
105$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{8}$$\\alpha^{14}$34[0, 1, 3, 4, 4, 11, 5, 62, 6, 170, 7, 304, 8, 475, 9, 650, 10, 732, 11, 660, 1, 3, 9, 10]
106$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{9}$$\\alpha^{10}$36[0, 1, 3, 6, 4, 11, 5, 55, 6, 168, 7, 310, 8, 485, 9, 655, 10, 712, 11, 650, 1, 3, 9, 11]
107$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{9}$$\\alpha^{11}$34[0, 1, 3, 4, 4, 12, 5, 62, 6, 162, 7, 304, 8, 498, 9, 650, 10, 702, 11, 660, 1, 3, 9, 12]
108$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{9}$$\\alpha^{12}$36[0, 1, 3, 6, 4, 17, 5, 57, 6, 151, 7, 300, 8, 495, 9, 675, 10, 723, 11, 630, 1, 3, 9, 13]
109$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{9}$$\\alpha^{13}$35[0, 1, 3, 5, 4, 15, 5, 57, 6, 155, 7, 309, 8, 497, 9, 657, 10, 715, 11, 647, 1, 3, 9, 14]
110$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{9}$$\\alpha^{14}$38[0, 1, 3, 8, 4, 13, 5, 52, 6, 162, 7, 300, 8, 485, 9, 680, 10, 732, 11, 640, 1, 3, 10, 11]
111$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{10}$$\\alpha^{11}$32[0, 1, 3, 2, 4, 18, 5, 61, 6, 161, 7, 300, 8, 470, 9, 683, 10, 735, 11, 626, 1, 3, 10, 12]
112$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{10}$$\\alpha^{12}$35[0, 1, 3, 5, 4, 14, 5, 63, 6, 157, 7, 293, 8, 502, 9, 663, 10, 695, 11, 663, 1, 3, 10, 13]
113$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{10}$$\\alpha^{13}$423[0, 1, 4, 23, 5, 56, 6, 154, 7, 320, 8, 479, 9, 664, 10, 700, 11, 640, 12, 537, 1, 3, 10, 14]
114$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{10}$$\\alpha^{14}$35[0, 1, 3, 5, 4, 13, 5, 60, 6, 160, 7, 301, 8, 495, 9, 660, 10, 712, 11, 655, 1, 3, 11, 12]
115$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{11}$$\\alpha^{12}$35[0, 1, 3, 5, 4, 11, 5, 61, 6, 173, 7, 303, 8, 467, 9, 643, 10, 735, 11, 683, 1, 3, 11, 13]
116$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{11}$$\\alpha^{13}$420[0, 1, 4, 20, 5, 64, 6, 150, 7, 326, 8, 472, 9, 618, 10, 764, 11, 668, 12, 472, 1, 3, 11, 14]
117$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{11}$$\\alpha^{14}$35[0, 1, 3, 5, 4, 14, 5, 58, 6, 160, 7, 303, 8, 488, 9, 672, 10, 720, 11, 627, 1, 3, 12, 13]
118$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{12}$$\\alpha^{13}$37[0, 1, 3, 7, 4, 14, 5, 55, 6, 157, 7, 303, 8, 502, 9, 663, 10, 695, 11, 653, 1, 3, 12, 14]
119$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{12}$$\\alpha^{14}$35[0, 1, 3, 5, 4, 14, 5, 58, 6, 158, 7, 309, 8, 490, 9, 650, 10, 732, 11, 655, 1, 3, 13, 14]
120$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{13}$$\\alpha^{14}$34[0, 1, 3, 4, 4, 13, 5, 67, 6, 165, 7, 288, 8, 477, 9, 663, 10, 735, 11, 668, 1, 4, 5, 6]
121$\\alpha^{1}$$\\alpha^{4}$$\\alpha^{5}$$\\alpha^{6}$37[0, 1, 3, 7, 4, 15, 5, 52, 6, 152, 7, 311, 8, 505, 9, 660, 10, 712, 11, 645, 1, 4, 5, 7]
122$\\alpha^{1}$$\\alpha^{4}$$\\alpha^{5}$$\\alpha^{7}$35[0, 1, 3, 5, 4, 11, 5, 61, 6, 173, 7, 303, 8, 467, 9, 643, 10, 735, 11, 683, 1, 4, 5, 8]
123$\\alpha^{1}$$\\alpha^{4}$$\\alpha^{5}$$\\alpha^{8}$39[0, 1, 3, 9, 4, 16, 5, 50, 6, 154, 7, 301, 8, 488, 9, 678, 10, 740, 11, 643, 1, 4, 5, 9]
124$\\alpha^{1}$$\\alpha^{4}$$\\alpha^{5}$$\\alpha^{9}$34[0, 1, 3, 4, 4, 14, 5, 59, 6, 157, 7, 312, 8, 500, 9, 647, 10, 705, 11, 652, 1, 4, 5, 10]
125$\\alpha^{1}$$\\alpha^{4}$$\\alpha^{5}$$\\alpha^{10}$310[0, 1, 3, 10, 4, 16, 5, 43, 6, 150, 7, 314, 8, 500, 9, 675, 10, 732, 11, 630, 1, 4, 5, 11]
126$\\alpha^{1}$$\\alpha^{4}$$\\alpha^{5}$$\\alpha^{11}$34[0, 1, 3, 4, 4, 14, 5, 59, 6, 158, 7, 312, 8, 498, 9, 647, 10, 700, 11, 652, 1, 4, 5, 12]
127$\\alpha^{1}$$\\alpha^{4}$$\\alpha^{5}$$\\alpha^{12}$35[0, 1, 3, 5, 4, 14, 5, 59, 6, 156, 7, 305, 8, 500, 9, 655, 10, 712, 11, 655, 1, 4, 5, 13]
128$\\alpha^{1}$$\\alpha^{4}$$\\alpha^{5}$$\\alpha^{13}$35[0, 1, 3, 5, 4, 12, 5, 58, 6, 162, 7, 309, 8, 500, 9, 650, 10, 692, 11, 655, 1, 4, 5, 14]
129$\\alpha^{1}$$\\alpha^{4}$$\\alpha^{5}$$\\alpha^{14}$36[0, 1, 3, 6, 4, 13, 5, 59, 6, 165, 7, 298, 8, 477, 9, 663, 10, 735, 11, 658, 1, 4, 6, 7]
130$\\alpha^{1}$$\\alpha^{4}$$\\alpha^{6}$$\\alpha^{7}$37[0, 1, 3, 7, 4, 13, 5, 53, 6, 160, 7, 305, 8, 495, 9, 675, 10, 712, 11, 625, 1, 4, 6, 8]
131$\\alpha^{1}$$\\alpha^{4}$$\\alpha^{6}$$\\alpha^{8}$38[0, 1, 3, 8, 4, 15, 5, 55, 6, 161, 7, 296, 8, 475, 9, 671, 10, 743, 11, 656, 1, 4, 6, 9]
132$\\alpha^{1}$$\\alpha^{4}$$\\alpha^{6}$$\\alpha^{9}$37[0, 1, 3, 7, 4, 14, 5, 51, 6, 156, 7, 315, 8, 500, 9, 655, 10, 712, 11, 645, 1, 4, 6, 10]
133$\\alpha^{1}$$\\alpha^{4}$$\\alpha^{6}$$\\alpha^{10}$37[0, 1, 3, 7, 4, 15, 5, 55, 6, 158, 7, 309, 8, 489, 9, 641, 10, 718, 11, 681, 1, 4, 6, 11]
134$\\alpha^{1}$$\\alpha^{4}$$\\alpha^{6}$$\\alpha^{11}$311[0, 1, 3, 11, 4, 20, 5, 44, 6, 142, 7, 307, 8, 496, 9, 676, 10, 748, 11, 641, 1, 4, 6, 12]
135$\\alpha^{1}$$\\alpha^{4}$$\\alpha^{6}$$\\alpha^{12}$312[0, 1, 3, 12, 4, 20, 5, 41, 6, 142, 7, 308, 8, 496, 9, 681, 10, 748, 11, 636, 1, 4, 6, 13]
136$\\alpha^{1}$$\\alpha^{4}$$\\alpha^{6}$$\\alpha^{13}$311[0, 1, 3, 11, 4, 20, 5, 48, 6, 146, 7, 295, 8, 484, 9, 684, 10, 756, 11, 649, 1, 4, 6, 14]
137$\\alpha^{1}$$\\alpha^{4}$$\\alpha^{6}$$\\alpha^{14}$37[0, 1, 3, 7, 4, 14, 5, 56, 6, 162, 7, 299, 8, 478, 9, 668, 10, 740, 11, 653, 1, 4, 7, 8]
138$\\alpha^{1}$$\\alpha^{4}$$\\alpha^{7}$$\\alpha^{8}$35[0, 1, 3, 5, 4, 13, 5, 63, 6, 166, 7, 293, 8, 473, 9, 663, 10, 740, 11, 663, 1, 4, 7, 9]
139$\\alpha^{1}$$\\alpha^{4}$$\\alpha^{7}$$\\alpha^{9}$38[0, 1, 3, 8, 4, 13, 5, 46, 6, 162, 7, 324, 8, 485, 9, 650, 10, 732, 11, 640, 1, 4, 7, 10]
140$\\alpha^{1}$$\\alpha^{4}$$\\alpha^{7}$$\\alpha^{10}$34[0, 1, 3, 4, 4, 17, 5, 63, 6, 152, 7, 300, 8, 491, 9, 655, 10, 728, 11, 660, 1, 4, 7, 11]
141$\\alpha^{1}$$\\alpha^{4}$$\\alpha^{7}$$\\alpha^{11}$35[0, 1, 3, 5, 4, 12, 5, 58, 6, 166, 7, 309, 8, 480, 9, 650, 10, 732, 11, 655, 1, 4, 7, 12]
142$\\alpha^{1}$$\\alpha^{4}$$\\alpha^{7}$$\\alpha^{12}$35[0, 1, 3, 5, 4, 15, 5, 58, 6, 151, 7, 311, 8, 509, 9, 640, 10, 707, 11, 675, 1, 4, 7, 13]
143$\\alpha^{1}$$\\alpha^{4}$$\\alpha^{7}$$\\alpha^{13}$34[0, 1, 3, 4, 4, 14, 5, 62, 6, 156, 7, 304, 8, 500, 9, 650, 10, 712, 11, 660, 1, 4, 7, 14]
144$\\alpha^{1}$$\\alpha^{4}$$\\alpha^{7}$$\\alpha^{14}$38[0, 1, 3, 8, 4, 12, 5, 53, 6, 172, 7, 304, 8, 464, 9, 661, 10, 738, 11, 656, 1, 4, 8, 9]
145$\\alpha^{1}$$\\alpha^{4}$$\\alpha^{8}$$\\alpha^{9}$35[0, 1, 3, 5, 4, 14, 5, 60, 6, 155, 7, 301, 8, 504, 9, 660, 10, 707, 11, 655, 1, 4, 8, 10]
146$\\alpha^{1}$$\\alpha^{4}$$\\alpha^{8}$$\\alpha^{10}$310[0, 1, 3, 10, 4, 18, 5, 49, 6, 152, 7, 302, 8, 482, 9, 669, 10, 746, 11, 654, 1, 4, 8, 11]
147$\\alpha^{1}$$\\alpha^{4}$$\\alpha^{8}$$\\alpha^{11}$36[0, 1, 3, 6, 4, 16, 5, 60, 6, 154, 7, 300, 8, 496, 9, 646, 10, 708, 11, 686, 1, 4, 8, 12]
148$\\alpha^{1}$$\\alpha^{4}$$\\alpha^{8}$$\\alpha^{12}$36[0, 1, 3, 6, 4, 16, 5, 52, 6, 151, 7, 316, 8, 502, 9, 662, 10, 715, 11, 622, 1, 4, 8, 13]
149$\\alpha^{1}$$\\alpha^{4}$$\\alpha^{8}$$\\alpha^{13}$35[0, 1, 3, 5, 4, 13, 5, 60, 6, 160, 7, 301, 8, 495, 9, 660, 10, 712, 11, 655, 1, 4, 8, 14]
150$\\alpha^{1}$$\\alpha^{4}$$\\alpha^{8}$$\\alpha^{14}$35[0, 1, 3, 5, 4, 16, 5, 62, 6, 154, 7, 297, 8, 488, 9, 658, 10, 740, 11, 663, 1, 4, 9, 10]
151$\\alpha^{1}$$\\alpha^{4}$$\\alpha^{9}$$\\alpha^{10}$36[0, 1, 3, 6, 4, 11, 5, 58, 6, 168, 7, 296, 8, 485, 9, 680, 10, 712, 11, 630, 1, 4, 9, 11]
152$\\alpha^{1}$$\\alpha^{4}$$\\alpha^{9}$$\\alpha^{11}$34[0, 1, 3, 4, 4, 10, 5, 61, 6, 170, 7, 310, 8, 488, 9, 635, 10, 702, 11, 680, 1, 4, 9, 12]
153$\\alpha^{1}$$\\alpha^{4}$$\\alpha^{9}$$\\alpha^{12}$36[0, 1, 3, 6, 4, 14, 5, 55, 6, 154, 7, 310, 8, 510, 9, 655, 10, 692, 11, 650, 1, 4, 9, 13]
154$\\alpha^{1}$$\\alpha^{4}$$\\alpha^{9}$$\\alpha^{13}$35[0, 1, 3, 5, 4, 14, 5, 54, 6, 157, 7, 321, 8, 500, 9, 642, 10, 705, 11, 647, 1, 4, 9, 14]
155$\\alpha^{1}$$\\alpha^{4}$$\\alpha^{9}$$\\alpha^{14}$38[0, 1, 3, 8, 4, 11, 5, 50, 6, 170, 7, 308, 8, 475, 9, 670, 10, 732, 11, 640, 1, 4, 10, 11]
156$\\alpha^{1}$$\\alpha^{4}$$\\alpha^{10}$$\\alpha^{11}$36[0, 1, 3, 6, 4, 14, 5, 60, 6, 164, 7, 298, 8, 476, 9, 656, 10, 728, 11, 666, 1, 4, 10, 12]
157$\\alpha^{1}$$\\alpha^{4}$$\\alpha^{10}$$\\alpha^{12}$35[0, 1, 3, 5, 4, 15, 5, 58, 6, 152, 7, 309, 8, 505, 9, 650, 10, 712, 11, 655, 1, 4, 10, 13]
158$\\alpha^{1}$$\\alpha^{4}$$\\alpha^{10}$$\\alpha^{13}$34[0, 1, 3, 4, 4, 18, 5, 59, 6, 144, 7, 314, 8, 508, 9, 637, 10, 720, 11, 672, 1, 4, 10, 14]
159$\\alpha^{1}$$\\alpha^{4}$$\\alpha^{10}$$\\alpha^{14}$35[0, 1, 3, 5, 4, 15, 5, 61, 6, 159, 7, 301, 8, 477, 9, 653, 10, 755, 11, 663, 1, 4, 11, 12]
160$\\alpha^{1}$$\\alpha^{4}$$\\alpha^{11}$$\\alpha^{12}$39[0, 1, 3, 9, 4, 16, 5, 50, 6, 154, 7, 301, 8, 488, 9, 678, 10, 740, 11, 643, 1, 4, 11, 13]
161$\\alpha^{1}$$\\alpha^{4}$$\\alpha^{11}$$\\alpha^{13}$38[0, 1, 3, 8, 4, 15, 5, 50, 6, 154, 7, 308, 8, 495, 9, 670, 10, 732, 11, 640, 1, 4, 11, 14]
162$\\alpha^{1}$$\\alpha^{4}$$\\alpha^{11}$$\\alpha^{14}$34[0, 1, 3, 4, 4, 14, 5, 60, 6, 164, 7, 308, 8, 468, 9, 652, 10, 760, 11, 652, 1, 4, 12, 13]
163$\\alpha^{1}$$\\alpha^{4}$$\\alpha^{12}$$\\alpha^{13}$39[0, 1, 3, 9, 4, 15, 5, 47, 6, 154, 7, 309, 8, 495, 9, 675, 10, 732, 11, 635, 1, 4, 12, 14]
164$\\alpha^{1}$$\\alpha^{4}$$\\alpha^{12}$$\\alpha^{14}$35[0, 1, 3, 5, 4, 15, 5, 56, 6, 155, 7, 311, 8, 497, 9, 662, 10, 715, 11, 627, 1, 4, 13, 14]
165$\\alpha^{1}$$\\alpha^{4}$$\\alpha^{13}$$\\alpha^{14}$36[0, 1, 3, 6, 4, 14, 5, 61, 6, 157, 7, 288, 8, 502, 9, 683, 10, 695, 11, 638, 1, 5, 6, 7]
166$\\alpha^{1}$$\\alpha^{5}$$\\alpha^{6}$$\\alpha^{7}$33[0, 1, 3, 3, 4, 15, 5, 68, 6, 149, 7, 291, 8, 519, 9, 660, 10, 687, 11, 665, 1, 5, 6, 8]
167$\\alpha^{1}$$\\alpha^{5}$$\\alpha^{6}$$\\alpha^{8}$32[0, 1, 3, 2, 4, 18, 5, 61, 6, 161, 7, 300, 8, 470, 9, 683, 10, 735, 11, 626, 1, 5, 6, 9]
168$\\alpha^{1}$$\\alpha^{5}$$\\alpha^{6}$$\\alpha^{9}$36[0, 1, 3, 6, 4, 14, 5, 60, 6, 164, 7, 298, 8, 476, 9, 656, 10, 728, 11, 666, 1, 5, 6, 10]
169$\\alpha^{1}$$\\alpha^{5}$$\\alpha^{6}$$\\alpha^{10}$36[0, 1, 3, 6, 4, 14, 5, 56, 6, 153, 7, 306, 8, 514, 9, 660, 10, 687, 11, 650, 1, 5, 6, 11]
170$\\alpha^{1}$$\\alpha^{5}$$\\alpha^{6}$$\\alpha^{11}$38[0, 1, 3, 8, 4, 18, 5, 55, 6, 146, 7, 296, 8, 506, 9, 671, 10, 708, 11, 656, 1, 5, 6, 12]
171$\\alpha^{1}$$\\alpha^{5}$$\\alpha^{6}$$\\alpha^{12}$32[0, 1, 3, 2, 4, 19, 5, 60, 6, 154, 7, 306, 8, 491, 9, 668, 10, 700, 11, 646, 1, 5, 6, 13]
172$\\alpha^{1}$$\\alpha^{5}$$\\alpha^{6}$$\\alpha^{13}$37[0, 1, 3, 7, 4, 14, 5, 58, 6, 164, 7, 295, 8, 474, 9, 666, 10, 738, 11, 661, 1, 5, 6, 14]
173$\\alpha^{1}$$\\alpha^{5}$$\\alpha^{6}$$\\alpha^{14}$34[0, 1, 3, 4, 4, 11, 5, 64, 6, 164, 7, 296, 8, 500, 9, 660, 10, 696, 11, 660, 1, 5, 7, 8]
174$\\alpha^{1}$$\\alpha^{5}$$\\alpha^{7}$$\\alpha^{8}$31[0, 1, 3, 1, 4, 18, 5, 66, 6, 148, 7, 315, 8, 496, 9, 624, 10, 728, 11, 679, 1, 5, 7, 9]
175$\\alpha^{1}$$\\alpha^{5}$$\\alpha^{7}$$\\alpha^{9}$35[0, 1, 3, 5, 4, 14, 5, 56, 6, 160, 7, 313, 8, 486, 9, 652, 10, 730, 11, 647, 1, 5, 7, 10]
176$\\alpha^{1}$$\\alpha^{5}$$\\alpha^{7}$$\\alpha^{10}$35[0, 1, 3, 5, 4, 15, 5, 58, 6, 158, 7, 303, 8, 483, 9, 672, 10, 740, 11, 627, 1, 5, 7, 11]
177$\\alpha^{1}$$\\alpha^{5}$$\\alpha^{7}$$\\alpha^{11}$31[0, 1, 3, 1, 4, 17, 5, 67, 6, 152, 7, 305, 8, 494, 9, 651, 10, 720, 11, 651, 1, 5, 7, 12]
178$\\alpha^{1}$$\\alpha^{5}$$\\alpha^{7}$$\\alpha^{12}$31[0, 1, 3, 1, 4, 17, 5, 66, 6, 153, 7, 309, 8, 493, 9, 646, 10, 711, 11, 651, 1, 5, 7, 13]
179$\\alpha^{1}$$\\alpha^{5}$$\\alpha^{7}$$\\alpha^{13}$36[0, 1, 3, 6, 4, 13, 5, 56, 6, 159, 7, 306, 8, 499, 9, 660, 10, 707, 11, 650, 1, 5, 7, 14]
180$\\alpha^{1}$$\\alpha^{5}$$\\alpha^{7}$$\\alpha^{14}$37[0, 1, 3, 7, 4, 14, 5, 55, 6, 162, 7, 305, 8, 478, 9, 653, 10, 740, 11, 673, 1, 5, 8, 9]
181$\\alpha^{1}$$\\alpha^{5}$$\\alpha^{8}$$\\alpha^{9}$419[0, 1, 4, 19, 5, 71, 6, 141, 7, 304, 8, 515, 9, 639, 10, 703, 11, 664, 12, 501, 1, 5, 8, 10]
182$\\alpha^{1}$$\\alpha^{5}$$\\alpha^{8}$$\\alpha^{10}$38[0, 1, 3, 8, 4, 13, 5, 50, 6, 162, 7, 308, 8, 485, 9, 670, 10, 732, 11, 640, 1, 5, 8, 11]
183$\\alpha^{1}$$\\alpha^{5}$$\\alpha^{8}$$\\alpha^{11}$31[0, 1, 3, 1, 4, 21, 5, 65, 6, 138, 7, 313, 8, 509, 9, 641, 10, 716, 11, 651, 1, 5, 8, 12]
184$\\alpha^{1}$$\\alpha^{5}$$\\alpha^{8}$$\\alpha^{12}$31[0, 1, 3, 1, 4, 17, 5, 67, 6, 155, 7, 305, 8, 483, 9, 651, 10, 731, 11, 651, 1, 5, 8, 13]
185$\\alpha^{1}$$\\alpha^{5}$$\\alpha^{8}$$\\alpha^{13}$31[0, 1, 3, 1, 4, 14, 5, 69, 6, 160, 7, 305, 8, 488, 9, 637, 10, 720, 11, 667, 1, 5, 8, 14]
186$\\alpha^{1}$$\\alpha^{5}$$\\alpha^{8}$$\\alpha^{14}$32[0, 1, 3, 2, 4, 17, 5, 67, 6, 154, 7, 296, 8, 481, 9, 669, 10, 748, 11, 634, 1, 5, 9, 10]
187$\\alpha^{1}$$\\alpha^{5}$$\\alpha^{9}$$\\alpha^{10}$36[0, 1, 3, 6, 4, 14, 5, 56, 6, 154, 7, 304, 8, 510, 9, 670, 10, 692, 11, 630, 1, 5, 9, 11]
188$\\alpha^{1}$$\\alpha^{5}$$\\alpha^{9}$$\\alpha^{11}$31[0, 1, 3, 1, 4, 23, 5, 56, 6, 146, 7, 331, 8, 479, 9, 630, 10, 748, 11, 663, 1, 5, 9, 12]
189$\\alpha^{1}$$\\alpha^{5}$$\\alpha^{9}$$\\alpha^{12}$31[0, 1, 3, 1, 4, 17, 5, 67, 6, 152, 7, 305, 8, 494, 9, 651, 10, 720, 11, 651, 1, 5, 9, 13]
190$\\alpha^{1}$$\\alpha^{5}$$\\alpha^{9}$$\\alpha^{13}$32[0, 1, 3, 2, 4, 21, 5, 49, 6, 164, 7, 326, 8, 459, 9, 669, 10, 720, 11, 630, 1, 5, 9, 14]
191$\\alpha^{1}$$\\alpha^{5}$$\\alpha^{9}$$\\alpha^{14}$32[0, 1, 3, 2, 4, 17, 5, 66, 6, 150, 7, 302, 8, 501, 9, 654, 10, 708, 11, 654, 1, 5, 10, 11]
192$\\alpha^{1}$$\\alpha^{5}$$\\alpha^{10}$$\\alpha^{11}$36[0, 1, 3, 6, 4, 14, 5, 56, 6, 153, 7, 306, 8, 514, 9, 660, 10, 687, 11, 650, 1, 5, 10, 12]
193$\\alpha^{1}$$\\alpha^{5}$$\\alpha^{10}$$\\alpha^{12}$34[0, 1, 3, 4, 4, 12, 5, 63, 6, 166, 7, 300, 8, 480, 9, 655, 10, 732, 11, 660, 1, 5, 10, 13]
194$\\alpha^{1}$$\\alpha^{5}$$\\alpha^{10}$$\\alpha^{13}$35[0, 1, 3, 5, 4, 12, 5, 60, 6, 161, 7, 299, 8, 504, 9, 670, 10, 687, 11, 635, 1, 5, 10, 14]
195$\\alpha^{1}$$\\alpha^{5}$$\\alpha^{10}$$\\alpha^{14}$37[0, 1, 3, 7, 4, 15, 5, 54, 6, 151, 7, 301, 8, 509, 9, 680, 10, 707, 11, 625, 1, 5, 11, 12]
196$\\alpha^{1}$$\\alpha^{5}$$\\alpha^{11}$$\\alpha^{12}$34[0, 1, 3, 4, 4, 14, 5, 59, 6, 157, 7, 312, 8, 500, 9, 647, 10, 705, 11, 652, 1, 5, 11, 13]
197$\\alpha^{1}$$\\alpha^{5}$$\\alpha^{11}$$\\alpha^{13}$31[0, 1, 3, 1, 4, 19, 5, 63, 6, 160, 7, 299, 8, 469, 9, 685, 10, 728, 11, 623, 1, 5, 11, 14]
198$\\alpha^{1}$$\\alpha^{5}$$\\alpha^{11}$$\\alpha^{14}$32[0, 1, 3, 2, 4, 22, 5, 53, 6, 147, 7, 330, 8, 488, 9, 645, 10, 723, 11, 638, 1, 5, 12, 13]
199$\\alpha^{1}$$\\alpha^{5}$$\\alpha^{12}$$\\alpha^{13}$35[0, 1, 3, 5, 4, 10, 5, 58, 6, 174, 7, 309, 8, 470, 9, 650, 10, 732, 11, 655, 1, 5, 12, 14]
200$\\alpha^{1}$$\\alpha^{5}$$\\alpha^{12}$$\\alpha^{14}$36[0, 1, 3, 6, 4, 12, 5, 58, 6, 164, 7, 298, 8, 490, 9, 670, 10, 712, 11, 650, 1, 5, 13, 14]
201$\\alpha^{1}$$\\alpha^{5}$$\\alpha^{13}$$\\alpha^{14}$33[0, 1, 3, 3, 4, 16, 5, 60, 6, 155, 7, 315, 8, 490, 9, 644, 10, 723, 11, 649, 1, 6, 7, 8]
202$\\alpha^{1}$$\\alpha^{6}$$\\alpha^{7}$$\\alpha^{8}$32[0, 1, 3, 2, 4, 17, 5, 70, 6, 148, 7, 288, 8, 503, 9, 672, 10, 720, 11, 642, 1, 6, 7, 9]
203$\\alpha^{1}$$\\alpha^{6}$$\\alpha^{7}$$\\alpha^{9}$36[0, 1, 3, 6, 4, 11, 5, 55, 6, 168, 7, 310, 8, 485, 9, 655, 10, 712, 11, 650, 1, 6, 7, 10]
204$\\alpha^{1}$$\\alpha^{6}$$\\alpha^{7}$$\\alpha^{10}$36[0, 1, 3, 6, 4, 11, 5, 58, 6, 168, 7, 296, 8, 485, 9, 680, 10, 712, 11, 630, 1, 6, 7, 11]
205$\\alpha^{1}$$\\alpha^{6}$$\\alpha^{7}$$\\alpha^{11}$36[0, 1, 3, 6, 4, 14, 5, 56, 6, 154, 7, 304, 8, 510, 9, 670, 10, 692, 11, 630, 1, 6, 7, 12]
206$\\alpha^{1}$$\\alpha^{6}$$\\alpha^{7}$$\\alpha^{12}$34[0, 1, 3, 4, 4, 14, 5, 59, 6, 159, 7, 312, 8, 492, 9, 647, 10, 715, 11, 652, 1, 6, 7, 13]
207$\\alpha^{1}$$\\alpha^{6}$$\\alpha^{7}$$\\alpha^{13}$36[0, 1, 3, 6, 4, 13, 5, 57, 6, 160, 7, 302, 8, 495, 9, 665, 10, 712, 11, 650, 1, 6, 7, 14]
208$\\alpha^{1}$$\\alpha^{6}$$\\alpha^{7}$$\\alpha^{14}$36[0, 1, 3, 6, 4, 10, 5, 59, 6, 173, 7, 292, 8, 474, 9, 685, 10, 727, 11, 630, 1, 6, 8, 9]
209$\\alpha^{1}$$\\alpha^{6}$$\\alpha^{8}$$\\alpha^{9}$32[0, 1, 3, 2, 4, 14, 5, 66, 6, 162, 7, 302, 8, 481, 9, 654, 10, 732, 11, 654, 1, 6, 8, 10]
210$\\alpha^{1}$$\\alpha^{6}$$\\alpha^{8}$$\\alpha^{10}$36[0, 1, 3, 6, 4, 12, 5, 62, 6, 176, 7, 290, 8, 446, 9, 666, 10, 768, 11, 666, 1, 6, 8, 11]
211$\\alpha^{1}$$\\alpha^{6}$$\\alpha^{8}$$\\alpha^{11}$37[0, 1, 3, 7, 4, 16, 5, 56, 6, 159, 7, 297, 8, 476, 9, 678, 10, 741, 11, 633, 1, 6, 8, 12]
212$\\alpha^{1}$$\\alpha^{6}$$\\alpha^{8}$$\\alpha^{12}$32[0, 1, 3, 2, 4, 18, 5, 58, 6, 162, 7, 312, 8, 466, 9, 668, 10, 740, 11, 626, 1, 6, 8, 13]
213$\\alpha^{1}$$\\alpha^{6}$$\\alpha^{8}$$\\alpha^{13}$36[0, 1, 3, 6, 4, 11, 5, 55, 6, 172, 7, 310, 8, 465, 9, 655, 10, 752, 11, 650, 1, 6, 8, 14]
214$\\alpha^{1}$$\\alpha^{6}$$\\alpha^{8}$$\\alpha^{14}$36[0, 1, 3, 6, 4, 12, 5, 57, 6, 166, 7, 302, 8, 480, 9, 665, 10, 732, 11, 650, 1, 6, 9, 10]
215$\\alpha^{1}$$\\alpha^{6}$$\\alpha^{9}$$\\alpha^{10}$34[0, 1, 3, 4, 4, 14, 5, 59, 6, 159, 7, 312, 8, 492, 9, 647, 10, 715, 11, 652, 1, 6, 9, 11]
216$\\alpha^{1}$$\\alpha^{6}$$\\alpha^{9}$$\\alpha^{11}$36[0, 1, 3, 6, 4, 14, 5, 61, 6, 163, 7, 294, 8, 480, 9, 661, 10, 723, 11, 666, 1, 6, 9, 12]
217$\\alpha^{1}$$\\alpha^{6}$$\\alpha^{9}$$\\alpha^{12}$33[0, 1, 3, 3, 4, 19, 5, 57, 6, 150, 7, 311, 8, 503, 9, 661, 10, 692, 11, 649, 1, 6, 9, 13]
218$\\alpha^{1}$$\\alpha^{6}$$\\alpha^{9}$$\\alpha^{13}$38[0, 1, 3, 8, 4, 15, 5, 56, 6, 162, 7, 292, 8, 471, 9, 676, 10, 748, 11, 656, 1, 6, 9, 14]
219$\\alpha^{1}$$\\alpha^{6}$$\\alpha^{9}$$\\alpha^{14}$32[0, 1, 3, 2, 4, 14, 5, 69, 6, 161, 7, 292, 8, 482, 9, 667, 10, 735, 11, 642, 1, 6, 10, 11]
220$\\alpha^{1}$$\\alpha^{6}$$\\alpha^{10}$$\\alpha^{11}$38[0, 1, 3, 8, 4, 18, 5, 55, 6, 146, 7, 296, 8, 506, 9, 671, 10, 708, 11, 656, 1, 6, 10, 12]
221$\\alpha^{1}$$\\alpha^{6}$$\\alpha^{10}$$\\alpha^{12}$33[0, 1, 3, 3, 4, 18, 5, 57, 6, 154, 7, 325, 8, 474, 9, 631, 10, 756, 11, 661, 1, 6, 10, 13]
222$\\alpha^{1}$$\\alpha^{6}$$\\alpha^{10}$$\\alpha^{13}$36[0, 1, 3, 6, 4, 15, 5, 56, 6, 151, 7, 306, 8, 509, 9, 660, 10, 707, 11, 650, 1, 6, 10, 14]
223$\\alpha^{1}$$\\alpha^{6}$$\\alpha^{10}$$\\alpha^{14}$33[0, 1, 3, 3, 4, 17, 5, 59, 6, 155, 7, 315, 8, 483, 9, 651, 10, 731, 11, 641, 1, 6, 11, 12]
224$\\alpha^{1}$$\\alpha^{6}$$\\alpha^{11}$$\\alpha^{12}$310[0, 1, 3, 10, 4, 16, 5, 43, 6, 150, 7, 314, 8, 500, 9, 675, 10, 732, 11, 630, 1, 6, 11, 13]
225$\\alpha^{1}$$\\alpha^{6}$$\\alpha^{11}$$\\alpha^{13}$310[0, 1, 3, 10, 4, 17, 5, 51, 6, 157, 7, 294, 8, 473, 9, 679, 10, 751, 11, 654, 1, 6, 11, 14]
226$\\alpha^{1}$$\\alpha^{6}$$\\alpha^{11}$$\\alpha^{14}$36[0, 1, 3, 6, 4, 13, 5, 56, 6, 160, 7, 308, 8, 495, 9, 650, 10, 712, 11, 670, 1, 6, 12, 13]
227$\\alpha^{1}$$\\alpha^{6}$$\\alpha^{12}$$\\alpha^{13}$310[0, 1, 3, 10, 4, 15, 5, 40, 6, 154, 7, 326, 8, 495, 9, 660, 10, 732, 11, 630, 1, 6, 12, 14]
228$\\alpha^{1}$$\\alpha^{6}$$\\alpha^{12}$$\\alpha^{14}$32[0, 1, 3, 2, 4, 16, 5, 63, 6, 154, 7, 314, 8, 496, 9, 639, 10, 708, 11, 654, 1, 6, 13, 14]
229$\\alpha^{1}$$\\alpha^{6}$$\\alpha^{13}$$\\alpha^{14}$36[0, 1, 3, 6, 4, 12, 5, 61, 6, 166, 7, 288, 8, 486, 9, 683, 10, 710, 11, 638, 1, 7, 8, 9]
230$\\alpha^{1}$$\\alpha^{7}$$\\alpha^{8}$$\\alpha^{9}$34[0, 1, 3, 4, 4, 15, 5, 67, 6, 154, 7, 288, 8, 501, 9, 663, 10, 710, 11, 668, 1, 7, 8, 10]
231$\\alpha^{1}$$\\alpha^{7}$$\\alpha^{8}$$\\alpha^{10}$34[0, 1, 3, 4, 4, 16, 5, 64, 6, 154, 7, 296, 8, 494, 9, 660, 10, 718, 11, 660, 1, 7, 8, 11]
232$\\alpha^{1}$$\\alpha^{7}$$\\alpha^{8}$$\\alpha^{11}$35[0, 1, 3, 5, 4, 14, 5, 60, 6, 155, 7, 301, 8, 504, 9, 660, 10, 707, 11, 655, 1, 7, 8, 12]
233$\\alpha^{1}$$\\alpha^{7}$$\\alpha^{8}$$\\alpha^{12}$419[0, 1, 4, 19, 5, 71, 6, 141, 7, 304, 8, 515, 9, 639, 10, 703, 11, 664, 12, 501, 1, 7, 8, 13]
234$\\alpha^{1}$$\\alpha^{7}$$\\alpha^{8}$$\\alpha^{13}$32[0, 1, 3, 2, 4, 14, 5, 66, 6, 162, 7, 302, 8, 481, 9, 654, 10, 732, 11, 654, 1, 7, 8, 14]
235$\\alpha^{1}$$\\alpha^{7}$$\\alpha^{8}$$\\alpha^{14}$34[0, 1, 3, 4, 4, 15, 5, 67, 6, 154, 7, 288, 8, 501, 9, 663, 10, 710, 11, 668, 1, 7, 9, 10]
236$\\alpha^{1}$$\\alpha^{7}$$\\alpha^{9}$$\\alpha^{10}$36[0, 1, 3, 6, 4, 13, 5, 57, 6, 160, 7, 302, 8, 495, 9, 665, 10, 712, 11, 650, 1, 7, 9, 11]
237$\\alpha^{1}$$\\alpha^{7}$$\\alpha^{9}$$\\alpha^{11}$34[0, 1, 3, 4, 4, 10, 5, 61, 6, 173, 7, 310, 8, 474, 9, 635, 10, 727, 11, 680, 1, 7, 9, 12]
238$\\alpha^{1}$$\\alpha^{7}$$\\alpha^{9}$$\\alpha^{12}$35[0, 1, 3, 5, 4, 12, 5, 63, 6, 168, 7, 291, 8, 476, 9, 673, 10, 730, 11, 643, 1, 7, 9, 13]
239$\\alpha^{1}$$\\alpha^{7}$$\\alpha^{9}$$\\alpha^{13}$35[0, 1, 3, 5, 4, 12, 5, 60, 6, 163, 7, 301, 8, 494, 9, 660, 10, 707, 11, 655, 1, 7, 9, 14]
240$\\alpha^{1}$$\\alpha^{7}$$\\alpha^{9}$$\\alpha^{14}$38[0, 1, 3, 8, 4, 11, 5, 50, 6, 170, 7, 308, 8, 475, 9, 670, 10, 732, 11, 640, 1, 7, 10, 11]
241$\\alpha^{1}$$\\alpha^{7}$$\\alpha^{10}$$\\alpha^{11}$32[0, 1, 3, 2, 4, 19, 5, 60, 6, 154, 7, 306, 8, 491, 9, 668, 10, 700, 11, 646, 1, 7, 10, 12]
242$\\alpha^{1}$$\\alpha^{7}$$\\alpha^{10}$$\\alpha^{12}$419[0, 1, 4, 19, 5, 68, 6, 148, 7, 308, 8, 492, 9, 648, 10, 728, 11, 648, 12, 492, 1, 7, 10, 13]
243$\\alpha^{1}$$\\alpha^{7}$$\\alpha^{10}$$\\alpha^{13}$424[0, 1, 4, 24, 5, 60, 6, 134, 7, 336, 8, 498, 9, 620, 10, 742, 11, 640, 12, 480, 1, 7, 10, 14]
244$\\alpha^{1}$$\\alpha^{7}$$\\alpha^{10}$$\\alpha^{14}$36[0, 1, 3, 6, 4, 17, 5, 54, 6, 152, 7, 316, 8, 491, 9, 640, 10, 728, 11, 670, 1, 7, 11, 12]
245$\\alpha^{1}$$\\alpha^{7}$$\\alpha^{11}$$\\alpha^{12}$34[0, 1, 3, 4, 4, 14, 5, 59, 6, 158, 7, 312, 8, 498, 9, 647, 10, 700, 11, 652, 1, 7, 11, 13]
246$\\alpha^{1}$$\\alpha^{7}$$\\alpha^{11}$$\\alpha^{13}$31[0, 1, 3, 1, 4, 17, 5, 67, 6, 150, 7, 309, 8, 501, 9, 639, 10, 708, 11, 659, 1, 7, 11, 14]
247$\\alpha^{1}$$\\alpha^{7}$$\\alpha^{11}$$\\alpha^{14}$34[0, 1, 3, 4, 4, 12, 5, 61, 6, 163, 7, 308, 8, 494, 9, 645, 10, 707, 11, 660, 1, 7, 12, 13]
248$\\alpha^{1}$$\\alpha^{7}$$\\alpha^{12}$$\\alpha^{13}$34[0, 1, 3, 4, 4, 12, 5, 59, 6, 167, 7, 312, 8, 482, 9, 647, 10, 715, 11, 652, 1, 7, 12, 14]
249$\\alpha^{1}$$\\alpha^{7}$$\\alpha^{12}$$\\alpha^{14}$34[0, 1, 3, 4, 4, 11, 5, 62, 6, 167, 7, 304, 8, 489, 9, 650, 10, 707, 11, 660, 1, 7, 13, 14]
250$\\alpha^{1}$$\\alpha^{7}$$\\alpha^{13}$$\\alpha^{14}$34[0, 1, 3, 4, 4, 15, 5, 64, 6, 150, 7, 296, 8, 515, 9, 660, 10, 692, 11, 660, 1, 8, 9, 10]
251$\\alpha^{1}$$\\alpha^{8}$$\\alpha^{9}$$\\alpha^{10}$36[0, 1, 3, 6, 4, 10, 5, 59, 6, 173, 7, 292, 8, 474, 9, 685, 10, 727, 11, 630, 1, 8, 9, 11]
252$\\alpha^{1}$$\\alpha^{8}$$\\alpha^{9}$$\\alpha^{11}$31[0, 1, 3, 1, 4, 18, 5, 65, 6, 150, 7, 319, 8, 486, 9, 619, 10, 748, 11, 679, 1, 8, 9, 12]
253$\\alpha^{1}$$\\alpha^{8}$$\\alpha^{9}$$\\alpha^{12}$35[0, 1, 3, 5, 4, 13, 5, 63, 6, 166, 7, 293, 8, 473, 9, 663, 10, 740, 11, 663, 1, 8, 9, 13]
254$\\alpha^{1}$$\\alpha^{8}$$\\alpha^{9}$$\\alpha^{13}$31[0, 1, 3, 1, 4, 18, 5, 66, 6, 148, 7, 315, 8, 496, 9, 624, 10, 728, 11, 679, 1, 8, 9, 14]
255$\\alpha^{1}$$\\alpha^{8}$$\\alpha^{9}$$\\alpha^{14}$32[0, 1, 3, 2, 4, 17, 5, 70, 6, 148, 7, 288, 8, 503, 9, 672, 10, 720, 11, 642, 1, 8, 10, 11]
256$\\alpha^{1}$$\\alpha^{8}$$\\alpha^{10}$$\\alpha^{11}$37[0, 1, 3, 7, 4, 14, 5, 58, 6, 164, 7, 295, 8, 474, 9, 666, 10, 738, 11, 661, 1, 8, 10, 12]
257$\\alpha^{1}$$\\alpha^{8}$$\\alpha^{10}$$\\alpha^{12}$34[0, 1, 3, 4, 4, 13, 5, 62, 6, 160, 7, 306, 8, 495, 9, 640, 10, 712, 11, 680, 1, 8, 10, 13]
258$\\alpha^{1}$$\\alpha^{8}$$\\alpha^{10}$$\\alpha^{13}$34[0, 1, 3, 4, 4, 13, 5, 63, 6, 160, 7, 300, 8, 495, 9, 655, 10, 712, 11, 660, 1, 8, 10, 14]
259$\\alpha^{1}$$\\alpha^{8}$$\\alpha^{10}$$\\alpha^{14}$35[0, 1, 3, 5, 4, 14, 5, 62, 6, 159, 7, 297, 8, 492, 9, 658, 10, 715, 11, 663, 1, 8, 11, 12]
260$\\alpha^{1}$$\\alpha^{8}$$\\alpha^{11}$$\\alpha^{12}$35[0, 1, 3, 5, 4, 14, 5, 59, 6, 156, 7, 305, 8, 500, 9, 655, 10, 712, 11, 655, 1, 8, 11, 13]
261$\\alpha^{1}$$\\alpha^{8}$$\\alpha^{11}$$\\alpha^{13}$31[0, 1, 3, 1, 4, 19, 5, 61, 6, 161, 7, 307, 8, 463, 9, 675, 10, 743, 11, 623, 1, 8, 11, 14]
262$\\alpha^{1}$$\\alpha^{8}$$\\alpha^{11}$$\\alpha^{14}$31[0, 1, 3, 1, 4, 22, 5, 57, 6, 162, 7, 303, 8, 454, 9, 699, 10, 740, 11, 615, 1, 8, 12, 13]
263$\\alpha^{1}$$\\alpha^{8}$$\\alpha^{12}$$\\alpha^{13}$34[0, 1, 3, 4, 4, 11, 5, 63, 6, 166, 7, 300, 8, 493, 9, 655, 10, 702, 11, 660, 1, 8, 12, 14]
264$\\alpha^{1}$$\\alpha^{8}$$\\alpha^{12}$$\\alpha^{14}$31[0, 1, 3, 1, 4, 15, 5, 71, 6, 156, 7, 293, 8, 496, 9, 659, 10, 712, 11, 659, 1, 8, 13, 14]
265$\\alpha^{1}$$\\alpha^{8}$$\\alpha^{13}$$\\alpha^{14}$418[0, 1, 4, 18, 5, 72, 6, 145, 7, 300, 8, 510, 9, 644, 10, 703, 11, 664, 12, 506, 1, 9, 10, 11]
266$\\alpha^{1}$$\\alpha^{9}$$\\alpha^{10}$$\\alpha^{11}$34[0, 1, 3, 4, 4, 11, 5, 64, 6, 164, 7, 296, 8, 500, 9, 660, 10, 696, 11, 660, 1, 9, 10, 12]
267$\\alpha^{1}$$\\alpha^{9}$$\\alpha^{10}$$\\alpha^{12}$33[0, 1, 3, 3, 4, 15, 5, 65, 6, 154, 7, 299, 8, 503, 9, 657, 10, 700, 11, 657, 1, 9, 10, 13]
268$\\alpha^{1}$$\\alpha^{9}$$\\alpha^{10}$$\\alpha^{13}$37[0, 1, 3, 7, 4, 13, 5, 53, 6, 160, 7, 305, 8, 495, 9, 675, 10, 712, 11, 625, 1, 9, 10, 14]
269$\\alpha^{1}$$\\alpha^{9}$$\\alpha^{10}$$\\alpha^{14}$33[0, 1, 3, 3, 4, 15, 5, 68, 6, 149, 7, 291, 8, 519, 9, 660, 10, 687, 11, 665, 1, 9, 11, 12]
270$\\alpha^{1}$$\\alpha^{9}$$\\alpha^{11}$$\\alpha^{12}$35[0, 1, 3, 5, 4, 12, 5, 58, 6, 162, 7, 309, 8, 500, 9, 650, 10, 692, 11, 655, 1, 9, 11, 13]
271$\\alpha^{1}$$\\alpha^{9}$$\\alpha^{11}$$\\alpha^{13}$31[0, 1, 3, 1, 4, 20, 5, 58, 6, 156, 7, 323, 8, 474, 9, 640, 10, 728, 11, 663, 1, 9, 11, 14]
272$\\alpha^{1}$$\\alpha^{9}$$\\alpha^{11}$$\\alpha^{14}$417[0, 1, 4, 17, 5, 70, 6, 151, 7, 308, 8, 495, 9, 634, 10, 723, 11, 664, 12, 491, 1, 9, 12, 13]
273$\\alpha^{1}$$\\alpha^{9}$$\\alpha^{12}$$\\alpha^{13}$36[0, 1, 3, 6, 4, 14, 5, 53, 6, 156, 7, 320, 8, 500, 9, 635, 10, 712, 11, 670, 1, 9, 12, 14]
274$\\alpha^{1}$$\\alpha^{9}$$\\alpha^{12}$$\\alpha^{14}$31[0, 1, 3, 1, 4, 17, 5, 71, 6, 153, 7, 291, 8, 485, 9, 669, 10, 743, 11, 639, 1, 9, 13, 14]
275$\\alpha^{1}$$\\alpha^{9}$$\\alpha^{13}$$\\alpha^{14}$31[0, 1, 3, 1, 4, 17, 5, 70, 6, 148, 7, 301, 8, 503, 9, 642, 10, 720, 11, 667, 1, 10, 11, 12]
276$\\alpha^{1}$$\\alpha^{10}$$\\alpha^{11}$$\\alpha^{12}$36[0, 1, 3, 6, 4, 13, 5, 59, 6, 165, 7, 298, 8, 477, 9, 663, 10, 735, 11, 658, 1, 10, 11, 13]
277$\\alpha^{1}$$\\alpha^{10}$$\\alpha^{11}$$\\alpha^{13}$32[0, 1, 3, 2, 4, 17, 5, 67, 6, 148, 7, 302, 8, 503, 9, 647, 10, 720, 11, 662, 1, 10, 11, 14]
278$\\alpha^{1}$$\\alpha^{10}$$\\alpha^{11}$$\\alpha^{14}$37[0, 1, 3, 7, 4, 15, 5, 52, 6, 152, 7, 311, 8, 505, 9, 660, 10, 712, 11, 645, 1, 10, 12, 13]
279$\\alpha^{1}$$\\alpha^{10}$$\\alpha^{12}$$\\alpha^{13}$34[0, 1, 3, 4, 4, 16, 5, 61, 6, 153, 7, 310, 8, 500, 9, 635, 10, 703, 11, 680, 1, 10, 12, 14]
280$\\alpha^{1}$$\\alpha^{10}$$\\alpha^{12}$$\\alpha^{14}$31[0, 1, 3, 1, 4, 20, 5, 64, 6, 140, 7, 317, 8, 512, 9, 636, 10, 706, 11, 651, 1, 10, 13, 14]
281$\\alpha^{1}$$\\alpha^{10}$$\\alpha^{13}$$\\alpha^{14}$31[0, 1, 3, 1, 4, 17, 5, 67, 6, 152, 7, 305, 8, 494, 9, 651, 10, 720, 11, 651, 1, 11, 12, 13]
282$\\alpha^{1}$$\\alpha^{11}$$\\alpha^{12}$$\\alpha^{13}$38[0, 1, 3, 8, 4, 13, 5, 49, 6, 165, 7, 316, 8, 477, 9, 653, 10, 735, 11, 648, 1, 11, 12, 14]
283$\\alpha^{1}$$\\alpha^{11}$$\\alpha^{12}$$\\alpha^{14}$34[0, 1, 3, 4, 4, 11, 5, 62, 6, 167, 7, 304, 8, 489, 9, 650, 10, 707, 11, 660, 1, 11, 13, 14]
284$\\alpha^{1}$$\\alpha^{11}$$\\alpha^{13}$$\\alpha^{14}$421[0, 1, 4, 21, 5, 63, 6, 152, 7, 312, 8, 479, 9, 655, 10, 728, 11, 648, 12, 503, 1, 12, 13, 14]
285$\\alpha^{1}$$\\alpha^{12}$$\\alpha^{13}$$\\alpha^{14}$34[0, 1, 3, 4, 4, 14, 5, 63, 6, 156, 7, 302, 8, 500, 9, 645, 10, 712, 11, 680, 2, 3, 4, 5]
286$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{4}$$\\alpha^{5}$421[0, 1, 4, 21, 5, 63, 6, 152, 7, 312, 8, 479, 9, 655, 10, 728, 11, 648, 12, 503, 2, 3, 4, 6]
287$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{4}$$\\alpha^{6}$35[0, 1, 3, 5, 4, 14, 5, 58, 6, 158, 7, 309, 8, 490, 9, 650, 10, 732, 11, 655, 2, 3, 4, 7]
288$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{4}$$\\alpha^{7}$35[0, 1, 3, 5, 4, 15, 5, 56, 6, 155, 7, 311, 8, 497, 9, 662, 10, 715, 11, 627, 2, 3, 4, 8]
289$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{4}$$\\alpha^{8}$36[0, 1, 3, 6, 4, 12, 5, 58, 6, 164, 7, 298, 8, 490, 9, 670, 10, 712, 11, 650, 2, 3, 4, 9]
290$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{4}$$\\alpha^{9}$32[0, 1, 3, 2, 4, 16, 5, 63, 6, 154, 7, 314, 8, 496, 9, 639, 10, 708, 11, 654, 2, 3, 4, 10]
291$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{4}$$\\alpha^{10}$34[0, 1, 3, 4, 4, 11, 5, 62, 6, 167, 7, 304, 8, 489, 9, 650, 10, 707, 11, 660, 2, 3, 4, 11]
292$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{4}$$\\alpha^{11}$31[0, 1, 3, 1, 4, 15, 5, 71, 6, 156, 7, 293, 8, 496, 9, 659, 10, 712, 11, 659, 2, 3, 4, 12]
293$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{4}$$\\alpha^{12}$31[0, 1, 3, 1, 4, 17, 5, 71, 6, 153, 7, 291, 8, 485, 9, 669, 10, 743, 11, 639, 2, 3, 4, 13]
294$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{4}$$\\alpha^{13}$31[0, 1, 3, 1, 4, 20, 5, 64, 6, 140, 7, 317, 8, 512, 9, 636, 10, 706, 11, 651, 2, 3, 4, 14]
295$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{4}$$\\alpha^{14}$34[0, 1, 3, 4, 4, 11, 5, 62, 6, 167, 7, 304, 8, 489, 9, 650, 10, 707, 11, 660, 2, 3, 5, 6]
296$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{5}$$\\alpha^{6}$34[0, 1, 3, 4, 4, 16, 5, 61, 6, 153, 7, 310, 8, 500, 9, 635, 10, 703, 11, 680, 2, 3, 5, 7]
297$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{5}$$\\alpha^{7}$31[0, 1, 3, 1, 4, 13, 5, 71, 6, 168, 7, 293, 8, 466, 9, 659, 10, 752, 11, 659, 2, 3, 5, 8]
298$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{5}$$\\alpha^{8}$36[0, 1, 3, 6, 4, 15, 5, 58, 6, 158, 7, 302, 8, 483, 9, 658, 10, 740, 11, 658, 2, 3, 5, 9]
299$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{5}$$\\alpha^{9}$31[0, 1, 3, 1, 4, 19, 5, 63, 6, 149, 7, 323, 8, 483, 9, 621, 10, 751, 11, 671, 2, 3, 5, 10]
300$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{5}$$\\alpha^{10}$34[0, 1, 3, 4, 4, 11, 5, 62, 6, 167, 7, 304, 8, 489, 9, 650, 10, 707, 11, 660, 2, 3, 5, 11]
301$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{5}$$\\alpha^{11}$35[0, 1, 3, 5, 4, 15, 5, 57, 6, 155, 7, 309, 8, 497, 9, 657, 10, 715, 11, 647, 2, 3, 5, 12]
302$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{5}$$\\alpha^{12}$423[0, 1, 4, 23, 5, 56, 6, 154, 7, 320, 8, 479, 9, 664, 10, 700, 11, 640, 12, 537, 2, 3, 5, 13]
303$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{5}$$\\alpha^{13}$420[0, 1, 4, 20, 5, 64, 6, 150, 7, 326, 8, 472, 9, 618, 10, 764, 11, 668, 12, 472, 2, 3, 5, 14]
304$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{5}$$\\alpha^{14}$37[0, 1, 3, 7, 4, 14, 5, 55, 6, 157, 7, 303, 8, 502, 9, 663, 10, 695, 11, 653, 2, 3, 6, 7]
305$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{6}$$\\alpha^{7}$35[0, 1, 3, 5, 4, 12, 5, 58, 6, 162, 7, 309, 8, 500, 9, 650, 10, 692, 11, 655, 2, 3, 6, 8]
306$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{6}$$\\alpha^{8}$311[0, 1, 3, 11, 4, 20, 5, 48, 6, 146, 7, 295, 8, 484, 9, 684, 10, 756, 11, 649, 2, 3, 6, 9]
307$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{6}$$\\alpha^{9}$34[0, 1, 3, 4, 4, 14, 5, 62, 6, 156, 7, 304, 8, 500, 9, 650, 10, 712, 11, 660, 2, 3, 6, 10]
308$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{6}$$\\alpha^{10}$35[0, 1, 3, 5, 4, 13, 5, 60, 6, 160, 7, 301, 8, 495, 9, 660, 10, 712, 11, 655, 2, 3, 6, 11]
309$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{6}$$\\alpha^{11}$35[0, 1, 3, 5, 4, 14, 5, 54, 6, 157, 7, 321, 8, 500, 9, 642, 10, 705, 11, 647, 2, 3, 6, 12]
310$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{6}$$\\alpha^{12}$34[0, 1, 3, 4, 4, 18, 5, 59, 6, 144, 7, 314, 8, 508, 9, 637, 10, 720, 11, 672, 2, 3, 6, 13]
311$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{6}$$\\alpha^{13}$38[0, 1, 3, 8, 4, 15, 5, 50, 6, 154, 7, 308, 8, 495, 9, 670, 10, 732, 11, 640, 2, 3, 6, 14]
312$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{6}$$\\alpha^{14}$39[0, 1, 3, 9, 4, 15, 5, 47, 6, 154, 7, 309, 8, 495, 9, 675, 10, 732, 11, 635, 2, 3, 7, 8]
313$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{7}$$\\alpha^{8}$37[0, 1, 3, 7, 4, 14, 5, 58, 6, 164, 7, 295, 8, 474, 9, 666, 10, 738, 11, 661, 2, 3, 7, 9]
314$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{7}$$\\alpha^{9}$36[0, 1, 3, 6, 4, 13, 5, 56, 6, 159, 7, 306, 8, 499, 9, 660, 10, 707, 11, 650, 2, 3, 7, 10]
315$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{7}$$\\alpha^{10}$31[0, 1, 3, 1, 4, 14, 5, 69, 6, 160, 7, 305, 8, 488, 9, 637, 10, 720, 11, 667, 2, 3, 7, 11]
316$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{7}$$\\alpha^{11}$32[0, 1, 3, 2, 4, 21, 5, 49, 6, 164, 7, 326, 8, 459, 9, 669, 10, 720, 11, 630, 2, 3, 7, 12]
317$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{7}$$\\alpha^{12}$35[0, 1, 3, 5, 4, 12, 5, 60, 6, 161, 7, 299, 8, 504, 9, 670, 10, 687, 11, 635, 2, 3, 7, 13]
318$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{7}$$\\alpha^{13}$31[0, 1, 3, 1, 4, 19, 5, 63, 6, 160, 7, 299, 8, 469, 9, 685, 10, 728, 11, 623, 2, 3, 7, 14]
319$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{7}$$\\alpha^{14}$35[0, 1, 3, 5, 4, 10, 5, 58, 6, 174, 7, 309, 8, 470, 9, 650, 10, 732, 11, 655, 2, 3, 8, 9]
320$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{8}$$\\alpha^{9}$36[0, 1, 3, 6, 4, 13, 5, 57, 6, 160, 7, 302, 8, 495, 9, 665, 10, 712, 11, 650, 2, 3, 8, 10]
321$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{8}$$\\alpha^{10}$36[0, 1, 3, 6, 4, 11, 5, 55, 6, 172, 7, 310, 8, 465, 9, 655, 10, 752, 11, 650, 2, 3, 8, 11]
322$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{8}$$\\alpha^{11}$38[0, 1, 3, 8, 4, 15, 5, 56, 6, 162, 7, 292, 8, 471, 9, 676, 10, 748, 11, 656, 2, 3, 8, 12]
323$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{8}$$\\alpha^{12}$36[0, 1, 3, 6, 4, 15, 5, 56, 6, 151, 7, 306, 8, 509, 9, 660, 10, 707, 11, 650, 2, 3, 8, 13]
324$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{8}$$\\alpha^{13}$310[0, 1, 3, 10, 4, 17, 5, 51, 6, 157, 7, 294, 8, 473, 9, 679, 10, 751, 11, 654, 2, 3, 8, 14]
325$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{8}$$\\alpha^{14}$310[0, 1, 3, 10, 4, 15, 5, 40, 6, 154, 7, 326, 8, 495, 9, 660, 10, 732, 11, 630, 2, 3, 9, 10]
326$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{9}$$\\alpha^{10}$32[0, 1, 3, 2, 4, 14, 5, 66, 6, 162, 7, 302, 8, 481, 9, 654, 10, 732, 11, 654, 2, 3, 9, 11]
327$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{9}$$\\alpha^{11}$35[0, 1, 3, 5, 4, 12, 5, 60, 6, 163, 7, 301, 8, 494, 9, 660, 10, 707, 11, 655, 2, 3, 9, 12]
328$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{9}$$\\alpha^{12}$424[0, 1, 4, 24, 5, 60, 6, 134, 7, 336, 8, 498, 9, 620, 10, 742, 11, 640, 12, 480, 2, 3, 9, 13]
329$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{9}$$\\alpha^{13}$31[0, 1, 3, 1, 4, 17, 5, 67, 6, 150, 7, 309, 8, 501, 9, 639, 10, 708, 11, 659, 2, 3, 9, 14]
330$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{9}$$\\alpha^{14}$34[0, 1, 3, 4, 4, 12, 5, 59, 6, 167, 7, 312, 8, 482, 9, 647, 10, 715, 11, 652, 2, 3, 10, 11]
331$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{10}$$\\alpha^{11}$31[0, 1, 3, 1, 4, 18, 5, 66, 6, 148, 7, 315, 8, 496, 9, 624, 10, 728, 11, 679, 2, 3, 10, 12]
332$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{10}$$\\alpha^{12}$34[0, 1, 3, 4, 4, 13, 5, 63, 6, 160, 7, 300, 8, 495, 9, 655, 10, 712, 11, 660, 2, 3, 10, 13]
333$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{10}$$\\alpha^{13}$31[0, 1, 3, 1, 4, 19, 5, 61, 6, 161, 7, 307, 8, 463, 9, 675, 10, 743, 11, 623, 2, 3, 10, 14]
334$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{10}$$\\alpha^{14}$34[0, 1, 3, 4, 4, 11, 5, 63, 6, 166, 7, 300, 8, 493, 9, 655, 10, 702, 11, 660, 2, 3, 11, 12]
335$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{11}$$\\alpha^{12}$37[0, 1, 3, 7, 4, 13, 5, 53, 6, 160, 7, 305, 8, 495, 9, 675, 10, 712, 11, 625, 2, 3, 11, 13]
336$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{11}$$\\alpha^{13}$31[0, 1, 3, 1, 4, 20, 5, 58, 6, 156, 7, 323, 8, 474, 9, 640, 10, 728, 11, 663, 2, 3, 11, 14]
337$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{11}$$\\alpha^{14}$36[0, 1, 3, 6, 4, 14, 5, 53, 6, 156, 7, 320, 8, 500, 9, 635, 10, 712, 11, 670, 2, 3, 12, 13]
338$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{12}$$\\alpha^{13}$32[0, 1, 3, 2, 4, 17, 5, 67, 6, 148, 7, 302, 8, 503, 9, 647, 10, 720, 11, 662, 2, 3, 12, 14]
339$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{12}$$\\alpha^{14}$34[0, 1, 3, 4, 4, 16, 5, 61, 6, 153, 7, 310, 8, 500, 9, 635, 10, 703, 11, 680, 2, 3, 13, 14]
340$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{13}$$\\alpha^{14}$38[0, 1, 3, 8, 4, 13, 5, 49, 6, 165, 7, 316, 8, 477, 9, 653, 10, 735, 11, 648, 2, 4, 5, 6]
341$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{5}$$\\alpha^{6}$31[0, 1, 3, 1, 4, 20, 5, 64, 6, 140, 7, 317, 8, 512, 9, 636, 10, 706, 11, 651, 2, 4, 5, 7]
342$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{5}$$\\alpha^{7}$420[0, 1, 4, 20, 5, 64, 6, 150, 7, 326, 8, 472, 9, 618, 10, 764, 11, 668, 12, 472, 2, 4, 5, 8]
343$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{5}$$\\alpha^{8}$38[0, 1, 3, 8, 4, 15, 5, 50, 6, 154, 7, 308, 8, 495, 9, 670, 10, 732, 11, 640, 2, 4, 5, 9]
344$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{5}$$\\alpha^{9}$31[0, 1, 3, 1, 4, 19, 5, 63, 6, 160, 7, 299, 8, 469, 9, 685, 10, 728, 11, 623, 2, 4, 5, 10]
345$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{5}$$\\alpha^{10}$310[0, 1, 3, 10, 4, 17, 5, 51, 6, 157, 7, 294, 8, 473, 9, 679, 10, 751, 11, 654, 2, 4, 5, 11]
346$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{5}$$\\alpha^{11}$31[0, 1, 3, 1, 4, 17, 5, 67, 6, 150, 7, 309, 8, 501, 9, 639, 10, 708, 11, 659, 2, 4, 5, 12]
347$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{5}$$\\alpha^{12}$31[0, 1, 3, 1, 4, 19, 5, 61, 6, 161, 7, 307, 8, 463, 9, 675, 10, 743, 11, 623, 2, 4, 5, 13]
348$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{5}$$\\alpha^{13}$31[0, 1, 3, 1, 4, 20, 5, 58, 6, 156, 7, 323, 8, 474, 9, 640, 10, 728, 11, 663, 2, 4, 5, 14]
349$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{5}$$\\alpha^{14}$32[0, 1, 3, 2, 4, 17, 5, 67, 6, 148, 7, 302, 8, 503, 9, 647, 10, 720, 11, 662, 2, 4, 6, 7]
350$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{6}$$\\alpha^{7}$31[0, 1, 3, 1, 4, 20, 5, 58, 6, 156, 7, 323, 8, 474, 9, 640, 10, 728, 11, 663, 2, 4, 6, 8]
351$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{6}$$\\alpha^{8}$36[0, 1, 3, 6, 4, 14, 5, 50, 6, 158, 7, 326, 8, 496, 9, 642, 10, 710, 11, 642, 2, 4, 6, 9]
352$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{6}$$\\alpha^{9}$31[0, 1, 3, 1, 4, 16, 5, 67, 6, 158, 7, 305, 8, 479, 9, 651, 10, 740, 11, 651, 2, 4, 6, 10]
353$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{6}$$\\alpha^{10}$35[0, 1, 3, 5, 4, 13, 5, 54, 6, 164, 7, 321, 8, 483, 9, 642, 10, 720, 11, 647, 2, 4, 6, 11]
354$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{6}$$\\alpha^{11}$32[0, 1, 3, 2, 4, 20, 5, 53, 6, 164, 7, 312, 8, 466, 9, 687, 10, 712, 11, 618, 2, 4, 6, 12]
355$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{6}$$\\alpha^{12}$36[0, 1, 3, 6, 4, 16, 5, 58, 6, 158, 7, 308, 8, 476, 9, 636, 10, 748, 11, 686, 2, 4, 6, 13]
356$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{6}$$\\alpha^{13}$36[0, 1, 3, 6, 4, 14, 5, 50, 6, 158, 7, 326, 8, 496, 9, 642, 10, 710, 11, 642, 2, 4, 6, 14]
357$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{6}$$\\alpha^{14}$32[0, 1, 3, 2, 4, 17, 5, 64, 6, 152, 7, 312, 8, 491, 9, 634, 10, 728, 11, 674, 2, 4, 7, 8]
358$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{7}$$\\alpha^{8}$34[0, 1, 3, 4, 4, 13, 5, 62, 6, 160, 7, 306, 8, 495, 9, 640, 10, 712, 11, 680, 2, 4, 7, 9]
359$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{7}$$\\alpha^{9}$34[0, 1, 3, 4, 4, 13, 5, 59, 6, 162, 7, 312, 8, 491, 9, 647, 10, 710, 11, 652, 2, 4, 7, 10]
360$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{7}$$\\alpha^{10}$34[0, 1, 3, 4, 4, 13, 5, 63, 6, 160, 7, 300, 8, 495, 9, 655, 10, 712, 11, 660, 2, 4, 7, 11]
361$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{7}$$\\alpha^{11}$32[0, 1, 3, 2, 4, 21, 5, 53, 6, 152, 7, 330, 8, 479, 9, 645, 10, 728, 11, 638, 2, 4, 7, 12]
362$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{7}$$\\alpha^{12}$35[0, 1, 3, 5, 4, 13, 5, 65, 6, 171, 7, 289, 8, 455, 9, 661, 10, 763, 11, 671, 2, 4, 7, 13]
363$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{7}$$\\alpha^{13}$31[0, 1, 3, 1, 4, 16, 5, 67, 6, 158, 7, 305, 8, 479, 9, 651, 10, 740, 11, 651, 2, 4, 7, 14]
364$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{7}$$\\alpha^{14}$417[0, 1, 4, 17, 5, 68, 6, 164, 7, 294, 8, 471, 9, 678, 10, 720, 11, 636, 12, 515, 2, 4, 8, 9]
365$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{8}$$\\alpha^{9}$34[0, 1, 3, 4, 4, 10, 5, 61, 6, 173, 7, 310, 8, 474, 9, 635, 10, 727, 11, 680, 2, 4, 8, 10]
366$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{8}$$\\alpha^{10}$39[0, 1, 3, 9, 4, 15, 5, 43, 6, 154, 7, 325, 8, 495, 9, 655, 10, 732, 11, 635, 2, 4, 8, 11]
367$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{8}$$\\alpha^{11}$36[0, 1, 3, 6, 4, 13, 5, 54, 6, 158, 7, 314, 8, 505, 9, 650, 10, 692, 11, 650, 2, 4, 8, 12]
368$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{8}$$\\alpha^{12}$35[0, 1, 3, 5, 4, 12, 5, 56, 6, 164, 7, 319, 8, 490, 9, 630, 10, 712, 11, 675, 2, 4, 8, 13]
369$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{8}$$\\alpha^{13}$35[0, 1, 3, 5, 4, 13, 5, 54, 6, 164, 7, 321, 8, 483, 9, 642, 10, 720, 11, 647, 2, 4, 8, 14]
370$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{8}$$\\alpha^{14}$34[0, 1, 3, 4, 4, 14, 5, 61, 6, 153, 7, 310, 8, 514, 9, 635, 10, 687, 11, 680, 2, 4, 9, 10]
371$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{9}$$\\alpha^{10}$36[0, 1, 3, 6, 4, 12, 5, 62, 6, 176, 7, 290, 8, 446, 9, 666, 10, 768, 11, 666, 2, 4, 9, 11]
372$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{9}$$\\alpha^{11}$35[0, 1, 3, 5, 4, 12, 5, 58, 6, 166, 7, 309, 8, 480, 9, 650, 10, 732, 11, 655, 2, 4, 9, 12]
373$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{9}$$\\alpha^{12}$31[0, 1, 3, 1, 4, 16, 5, 67, 6, 158, 7, 311, 8, 476, 9, 629, 10, 748, 11, 679, 2, 4, 9, 13]
374$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{9}$$\\alpha^{13}$32[0, 1, 3, 2, 4, 20, 5, 53, 6, 164, 7, 312, 8, 466, 9, 687, 10, 712, 11, 618, 2, 4, 9, 14]
375$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{9}$$\\alpha^{14}$34[0, 1, 3, 4, 4, 11, 5, 63, 6, 170, 7, 298, 8, 475, 9, 665, 10, 732, 11, 640, 2, 4, 10, 11]
376$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{10}$$\\alpha^{11}$35[0, 1, 3, 5, 4, 14, 5, 56, 6, 160, 7, 313, 8, 486, 9, 652, 10, 730, 11, 647, 2, 4, 10, 12]
377$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{10}$$\\alpha^{12}$35[0, 1, 3, 5, 4, 12, 5, 59, 6, 165, 7, 305, 8, 484, 9, 655, 10, 727, 11, 655, 2, 4, 10, 13]
378$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{10}$$\\alpha^{13}$36[0, 1, 3, 6, 4, 16, 5, 58, 6, 158, 7, 308, 8, 476, 9, 636, 10, 748, 11, 686, 2, 4, 10, 14]
379$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{10}$$\\alpha^{14}$34[0, 1, 3, 4, 4, 13, 5, 62, 6, 158, 7, 304, 8, 505, 9, 650, 10, 692, 11, 660, 2, 4, 11, 12]
380$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{11}$$\\alpha^{12}$38[0, 1, 3, 8, 4, 15, 5, 55, 6, 161, 7, 296, 8, 475, 9, 671, 10, 743, 11, 656, 2, 4, 11, 13]
381$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{11}$$\\alpha^{13}$36[0, 1, 3, 6, 4, 14, 5, 50, 6, 158, 7, 326, 8, 496, 9, 642, 10, 710, 11, 642, 2, 4, 11, 14]
382$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{11}$$\\alpha^{14}$31[0, 1, 3, 1, 4, 16, 5, 69, 6, 152, 7, 305, 8, 498, 9, 637, 10, 720, 11, 667, 2, 4, 12, 13]
383$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{12}$$\\alpha^{13}$32[0, 1, 3, 2, 4, 17, 5, 64, 6, 152, 7, 312, 8, 491, 9, 634, 10, 728, 11, 674, 2, 4, 12, 14]
384$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{12}$$\\alpha^{14}$31[0, 1, 3, 1, 4, 13, 5, 71, 6, 168, 7, 293, 8, 466, 9, 659, 10, 752, 11, 659, 2, 4, 13, 14]
385$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{13}$$\\alpha^{14}$37[0, 1, 3, 7, 4, 14, 5, 52, 6, 155, 7, 311, 8, 504, 9, 660, 10, 707, 11, 645, 2, 5, 6, 7]
386$\\alpha^{2}$$\\alpha^{5}$$\\alpha^{6}$$\\alpha^{7}$417[0, 1, 4, 17, 5, 70, 6, 151, 7, 308, 8, 495, 9, 634, 10, 723, 11, 664, 12, 491, 2, 5, 6, 8]
387$\\alpha^{2}$$\\alpha^{5}$$\\alpha^{6}$$\\alpha^{8}$35[0, 1, 3, 5, 4, 14, 5, 63, 6, 157, 7, 293, 8, 502, 9, 663, 10, 695, 11, 663, 2, 5, 6, 9]
388$\\alpha^{2}$$\\alpha^{5}$$\\alpha^{6}$$\\alpha^{9}$35[0, 1, 3, 5, 4, 15, 5, 58, 6, 152, 7, 309, 8, 505, 9, 650, 10, 712, 11, 655, 2, 5, 6, 10]
389$\\alpha^{2}$$\\alpha^{5}$$\\alpha^{6}$$\\alpha^{10}$34[0, 1, 3, 4, 4, 12, 5, 63, 6, 166, 7, 300, 8, 480, 9, 655, 10, 732, 11, 660, 2, 5, 6, 11]
390$\\alpha^{2}$$\\alpha^{5}$$\\alpha^{6}$$\\alpha^{11}$33[0, 1, 3, 3, 4, 18, 5, 57, 6, 154, 7, 325, 8, 474, 9, 631, 10, 756, 11, 661, 2, 5, 6, 12]
391$\\alpha^{2}$$\\alpha^{5}$$\\alpha^{6}$$\\alpha^{12}$419[0, 1, 4, 19, 5, 68, 6, 148, 7, 308, 8, 492, 9, 648, 10, 728, 11, 648, 12, 492, 2, 5, 6, 13]
392$\\alpha^{2}$$\\alpha^{5}$$\\alpha^{6}$$\\alpha^{13}$34[0, 1, 3, 4, 4, 13, 5, 62, 6, 160, 7, 306, 8, 495, 9, 640, 10, 712, 11, 680, 2, 5, 6, 14]
393$\\alpha^{2}$$\\alpha^{5}$$\\alpha^{6}$$\\alpha^{14}$33[0, 1, 3, 3, 4, 15, 5, 65, 6, 154, 7, 299, 8, 503, 9, 657, 10, 700, 11, 657, 2, 5, 7, 8]
394$\\alpha^{2}$$\\alpha^{5}$$\\alpha^{7}$$\\alpha^{8}$34[0, 1, 3, 4, 4, 13, 5, 63, 6, 160, 7, 300, 8, 495, 9, 655, 10, 712, 11, 660, 2, 5, 7, 9]
395$\\alpha^{2}$$\\alpha^{5}$$\\alpha^{7}$$\\alpha^{9}$35[0, 1, 3, 5, 4, 12, 5, 59, 6, 165, 7, 305, 8, 484, 9, 655, 10, 727, 11, 655, 2, 5, 7, 10]
396$\\alpha^{2}$$\\alpha^{5}$$\\alpha^{7}$$\\alpha^{10}$34[0, 1, 3, 4, 4, 11, 5, 62, 6, 170, 7, 304, 8, 475, 9, 650, 10, 732, 11, 660, 2, 5, 7, 11]
397$\\alpha^{2}$$\\alpha^{5}$$\\alpha^{7}$$\\alpha^{11}$31[0, 1, 3, 1, 4, 21, 5, 60, 6, 146, 7, 331, 8, 477, 9, 618, 10, 764, 11, 663, 2, 5, 7, 12]
398$\\alpha^{2}$$\\alpha^{5}$$\\alpha^{7}$$\\alpha^{12}$34[0, 1, 3, 4, 4, 11, 5, 61, 6, 173, 7, 304, 8, 467, 9, 657, 10, 735, 11, 652, 2, 5, 7, 13]
399$\\alpha^{2}$$\\alpha^{5}$$\\alpha^{7}$$\\alpha^{13}$34[0, 1, 3, 4, 4, 13, 5, 59, 6, 162, 7, 312, 8, 491, 9, 647, 10, 710, 11, 652, 2, 5, 7, 14]
400$\\alpha^{2}$$\\alpha^{5}$$\\alpha^{7}$$\\alpha^{14}$32[0, 1, 3, 2, 4, 18, 5, 57, 6, 165, 7, 312, 8, 458, 9, 675, 10, 743, 11, 618, 2, 5, 8, 9]
401$\\alpha^{2}$$\\alpha^{5}$$\\alpha^{8}$$\\alpha^{9}$35[0, 1, 3, 5, 4, 12, 5, 63, 6, 168, 7, 291, 8, 476, 9, 673, 10, 730, 11, 643, 2, 5, 8, 10]
402$\\alpha^{2}$$\\alpha^{5}$$\\alpha^{8}$$\\alpha^{10}$38[0, 1, 3, 8, 4, 14, 5, 49, 6, 158, 7, 312, 8, 490, 9, 665, 10, 732, 11, 640, 2, 5, 8, 11]
403$\\alpha^{2}$$\\alpha^{5}$$\\alpha^{8}$$\\alpha^{11}$36[0, 1, 3, 6, 4, 16, 5, 58, 6, 152, 7, 302, 8, 498, 9, 658, 10, 720, 11, 658, 2, 5, 8, 12]
404$\\alpha^{2}$$\\alpha^{5}$$\\alpha^{8}$$\\alpha^{12}$35[0, 1, 3, 5, 4, 14, 5, 57, 6, 161, 7, 309, 8, 482, 9, 657, 10, 735, 11, 647, 2, 5, 8, 13]
405$\\alpha^{2}$$\\alpha^{5}$$\\alpha^{8}$$\\alpha^{13}$34[0, 1, 3, 4, 4, 13, 5, 63, 6, 160, 7, 300, 8, 495, 9, 655, 10, 712, 11, 660, 2, 5, 8, 14]
406$\\alpha^{2}$$\\alpha^{5}$$\\alpha^{8}$$\\alpha^{14}$36[0, 1, 3, 6, 4, 14, 5, 55, 6, 158, 7, 310, 8, 490, 9, 655, 10, 732, 11, 650, 2, 5, 9, 10]
407$\\alpha^{2}$$\\alpha^{5}$$\\alpha^{9}$$\\alpha^{10}$37[0, 1, 3, 7, 4, 16, 5, 56, 6, 159, 7, 297, 8, 476, 9, 678, 10, 741, 11, 633, 2, 5, 9, 11]
408$\\alpha^{2}$$\\alpha^{5}$$\\alpha^{9}$$\\alpha^{11}$37[0, 1, 3, 7, 4, 15, 5, 53, 6, 157, 7, 313, 8, 487, 9, 643, 10, 735, 11, 673, 2, 5, 9, 12]
409$\\alpha^{2}$$\\alpha^{5}$$\\alpha^{9}$$\\alpha^{12}$31[0, 1, 3, 1, 4, 22, 5, 59, 6, 142, 7, 335, 8, 482, 9, 613, 10, 764, 11, 663, 2, 5, 9, 13]
410$\\alpha^{2}$$\\alpha^{5}$$\\alpha^{9}$$\\alpha^{13}$32[0, 1, 3, 2, 4, 21, 5, 53, 6, 152, 7, 330, 8, 479, 9, 645, 10, 728, 11, 638, 2, 5, 9, 14]
411$\\alpha^{2}$$\\alpha^{5}$$\\alpha^{9}$$\\alpha^{14}$33[0, 1, 3, 3, 4, 13, 5, 65, 6, 160, 7, 299, 8, 498, 9, 657, 10, 704, 11, 657, 2, 5, 10, 11]
412$\\alpha^{2}$$\\alpha^{5}$$\\alpha^{10}$$\\alpha^{11}$35[0, 1, 3, 5, 4, 15, 5, 58, 6, 158, 7, 303, 8, 483, 9, 672, 10, 740, 11, 627, 2, 5, 10, 12]
413$\\alpha^{2}$$\\alpha^{5}$$\\alpha^{10}$$\\alpha^{12}$34[0, 1, 3, 4, 4, 11, 5, 62, 6, 170, 7, 304, 8, 475, 9, 650, 10, 732, 11, 660, 2, 5, 10, 13]
414$\\alpha^{2}$$\\alpha^{5}$$\\alpha^{10}$$\\alpha^{13}$35[0, 1, 3, 5, 4, 13, 5, 65, 6, 171, 7, 289, 8, 455, 9, 661, 10, 763, 11, 671, 2, 5, 10, 14]
415$\\alpha^{2}$$\\alpha^{5}$$\\alpha^{10}$$\\alpha^{14}$36[0, 1, 3, 6, 4, 14, 5, 57, 6, 156, 7, 300, 8, 500, 9, 675, 10, 712, 11, 630, 2, 5, 11, 12]
416$\\alpha^{2}$$\\alpha^{5}$$\\alpha^{11}$$\\alpha^{12}$37[0, 1, 3, 7, 4, 14, 5, 51, 6, 156, 7, 315, 8, 500, 9, 655, 10, 712, 11, 645, 2, 5, 11, 13]
417$\\alpha^{2}$$\\alpha^{5}$$\\alpha^{11}$$\\alpha^{13}$31[0, 1, 3, 1, 4, 16, 5, 67, 6, 158, 7, 305, 8, 479, 9, 651, 10, 740, 11, 651, 2, 5, 11, 14]
418$\\alpha^{2}$$\\alpha^{5}$$\\alpha^{11}$$\\alpha^{14}$33[0, 1, 3, 3, 4, 19, 5, 58, 6, 145, 7, 319, 8, 503, 9, 646, 10, 711, 11, 641, 2, 5, 12, 13]
419$\\alpha^{2}$$\\alpha^{5}$$\\alpha^{12}$$\\alpha^{13}$417[0, 1, 4, 17, 5, 68, 6, 164, 7, 294, 8, 471, 9, 678, 10, 720, 11, 636, 12, 515, 2, 5, 12, 14]
420$\\alpha^{2}$$\\alpha^{5}$$\\alpha^{12}$$\\alpha^{14}$36[0, 1, 3, 6, 4, 15, 5, 58, 6, 158, 7, 302, 8, 483, 9, 658, 10, 740, 11, 658, 2, 5, 13, 14]
421$\\alpha^{2}$$\\alpha^{5}$$\\alpha^{13}$$\\alpha^{14}$36[0, 1, 3, 6, 4, 10, 5, 58, 6, 178, 7, 302, 8, 456, 9, 658, 10, 750, 11, 658, 2, 6, 7, 8]
422$\\alpha^{2}$$\\alpha^{6}$$\\alpha^{7}$$\\alpha^{8}$35[0, 1, 3, 5, 4, 14, 5, 62, 6, 159, 7, 297, 8, 492, 9, 658, 10, 715, 11, 663, 2, 6, 7, 9]
423$\\alpha^{2}$$\\alpha^{6}$$\\alpha^{7}$$\\alpha^{9}$34[0, 1, 3, 4, 4, 12, 5, 62, 6, 162, 7, 304, 8, 498, 9, 650, 10, 702, 11, 660, 2, 6, 7, 10]
424$\\alpha^{2}$$\\alpha^{6}$$\\alpha^{7}$$\\alpha^{10}$34[0, 1, 3, 4, 4, 10, 5, 61, 6, 170, 7, 310, 8, 488, 9, 635, 10, 702, 11, 680, 2, 6, 7, 11]
425$\\alpha^{2}$$\\alpha^{6}$$\\alpha^{7}$$\\alpha^{11}$31[0, 1, 3, 1, 4, 23, 5, 56, 6, 146, 7, 331, 8, 479, 9, 630, 10, 748, 11, 663, 2, 6, 7, 12]
426$\\alpha^{2}$$\\alpha^{6}$$\\alpha^{7}$$\\alpha^{12}$36[0, 1, 3, 6, 4, 14, 5, 61, 6, 163, 7, 294, 8, 480, 9, 661, 10, 723, 11, 666, 2, 6, 7, 13]
427$\\alpha^{2}$$\\alpha^{6}$$\\alpha^{7}$$\\alpha^{13}$34[0, 1, 3, 4, 4, 10, 5, 61, 6, 173, 7, 310, 8, 474, 9, 635, 10, 727, 11, 680, 2, 6, 7, 14]
428$\\alpha^{2}$$\\alpha^{6}$$\\alpha^{7}$$\\alpha^{14}$31[0, 1, 3, 1, 4, 18, 5, 65, 6, 150, 7, 319, 8, 486, 9, 619, 10, 748, 11, 679, 2, 6, 8, 9]
429$\\alpha^{2}$$\\alpha^{6}$$\\alpha^{8}$$\\alpha^{9}$35[0, 1, 3, 5, 4, 12, 5, 60, 6, 163, 7, 301, 8, 494, 9, 660, 10, 707, 11, 655, 2, 6, 8, 10]
430$\\alpha^{2}$$\\alpha^{6}$$\\alpha^{8}$$\\alpha^{10}$35[0, 1, 3, 5, 4, 12, 5, 58, 6, 166, 7, 309, 8, 480, 9, 650, 10, 732, 11, 655, 2, 6, 8, 11]
431$\\alpha^{2}$$\\alpha^{6}$$\\alpha^{8}$$\\alpha^{11}$37[0, 1, 3, 7, 4, 15, 5, 53, 6, 157, 7, 313, 8, 487, 9, 643, 10, 735, 11, 673, 2, 6, 8, 12]
432$\\alpha^{2}$$\\alpha^{6}$$\\alpha^{8}$$\\alpha^{12}$35[0, 1, 3, 5, 4, 14, 5, 55, 6, 160, 7, 317, 8, 488, 9, 647, 10, 720, 11, 647, 2, 6, 8, 13]
433$\\alpha^{2}$$\\alpha^{6}$$\\alpha^{8}$$\\alpha^{13}$39[0, 1, 3, 9, 4, 15, 5, 43, 6, 154, 7, 325, 8, 495, 9, 655, 10, 732, 11, 635, 2, 6, 8, 14]
434$\\alpha^{2}$$\\alpha^{6}$$\\alpha^{8}$$\\alpha^{14}$310[0, 1, 3, 10, 4, 16, 5, 45, 6, 154, 7, 310, 8, 488, 9, 673, 10, 740, 11, 638, 2, 6, 9, 10]
435$\\alpha^{2}$$\\alpha^{6}$$\\alpha^{9}$$\\alpha^{10}$32[0, 1, 3, 2, 4, 18, 5, 58, 6, 162, 7, 312, 8, 466, 9, 668, 10, 740, 11, 626, 2, 6, 9, 11]
436$\\alpha^{2}$$\\alpha^{6}$$\\alpha^{9}$$\\alpha^{11}$35[0, 1, 3, 5, 4, 14, 5, 55, 6, 160, 7, 317, 8, 488, 9, 647, 10, 720, 11, 647, 2, 6, 9, 12]
437$\\alpha^{2}$$\\alpha^{6}$$\\alpha^{9}$$\\alpha^{12}$423[0, 1, 4, 23, 5, 61, 6, 145, 7, 316, 8, 491, 9, 657, 10, 711, 11, 640, 12, 521, 2, 6, 9, 13]
438$\\alpha^{2}$$\\alpha^{6}$$\\alpha^{9}$$\\alpha^{13}$36[0, 1, 3, 6, 4, 13, 5, 54, 6, 158, 7, 314, 8, 505, 9, 650, 10, 692, 11, 650, 2, 6, 9, 14]
439$\\alpha^{2}$$\\alpha^{6}$$\\alpha^{9}$$\\alpha^{14}$31[0, 1, 3, 1, 4, 20, 5, 62, 6, 152, 7, 311, 8, 486, 9, 648, 10, 720, 11, 671, 2, 6, 10, 11]
440$\\alpha^{2}$$\\alpha^{6}$$\\alpha^{10}$$\\alpha^{11}$31[0, 1, 3, 1, 4, 17, 5, 67, 6, 152, 7, 305, 8, 494, 9, 651, 10, 720, 11, 651, 2, 6, 10, 12]
441$\\alpha^{2}$$\\alpha^{6}$$\\alpha^{10}$$\\alpha^{12}$31[0, 1, 3, 1, 4, 21, 5, 60, 6, 146, 7, 331, 8, 477, 9, 618, 10, 764, 11, 663, 2, 6, 10, 13]
442$\\alpha^{2}$$\\alpha^{6}$$\\alpha^{10}$$\\alpha^{13}$35[0, 1, 3, 5, 4, 12, 5, 56, 6, 164, 7, 319, 8, 490, 9, 630, 10, 712, 11, 675, 2, 6, 10, 14]
443$\\alpha^{2}$$\\alpha^{6}$$\\alpha^{10}$$\\alpha^{14}$31[0, 1, 3, 1, 4, 17, 5, 62, 6, 161, 7, 309, 8, 485, 9, 658, 10, 695, 11, 651, 2, 6, 11, 12]
444$\\alpha^{2}$$\\alpha^{6}$$\\alpha^{11}$$\\alpha^{12}$37[0, 1, 3, 7, 4, 15, 5, 55, 6, 158, 7, 309, 8, 489, 9, 641, 10, 718, 11, 681, 2, 6, 11, 13]
445$\\alpha^{2}$$\\alpha^{6}$$\\alpha^{11}$$\\alpha^{13}$35[0, 1, 3, 5, 4, 13, 5, 54, 6, 164, 7, 321, 8, 483, 9, 642, 10, 720, 11, 647, 2, 6, 11, 14]
446$\\alpha^{2}$$\\alpha^{6}$$\\alpha^{11}$$\\alpha^{14}$32[0, 1, 3, 2, 4, 16, 5, 63, 6, 155, 7, 314, 8, 490, 9, 639, 10, 723, 11, 654, 2, 6, 12, 13]
447$\\alpha^{2}$$\\alpha^{6}$$\\alpha^{12}$$\\alpha^{13}$34[0, 1, 3, 4, 4, 14, 5, 61, 6, 153, 7, 310, 8, 514, 9, 635, 10, 687, 11, 680, 2, 6, 12, 14]
448$\\alpha^{2}$$\\alpha^{6}$$\\alpha^{12}$$\\alpha^{14}$31[0, 1, 3, 1, 4, 19, 5, 63, 6, 149, 7, 323, 8, 483, 9, 621, 10, 751, 11, 671, 2, 6, 13, 14]
449$\\alpha^{2}$$\\alpha^{6}$$\\alpha^{13}$$\\alpha^{14}$39[0, 1, 3, 9, 4, 12, 5, 46, 6, 166, 7, 313, 8, 480, 9, 670, 10, 732, 11, 635, 2, 7, 8, 9]
450$\\alpha^{2}$$\\alpha^{7}$$\\alpha^{8}$$\\alpha^{9}$38[0, 1, 3, 8, 4, 11, 5, 50, 6, 170, 7, 308, 8, 475, 9, 670, 10, 732, 11, 640, 2, 7, 8, 10]
451$\\alpha^{2}$$\\alpha^{7}$$\\alpha^{8}$$\\alpha^{10}$34[0, 1, 3, 4, 4, 10, 5, 63, 6, 172, 7, 298, 8, 480, 9, 665, 10, 712, 11, 640, 2, 7, 8, 11]
452$\\alpha^{2}$$\\alpha^{7}$$\\alpha^{8}$$\\alpha^{11}$310[0, 1, 3, 10, 4, 18, 5, 49, 6, 152, 7, 302, 8, 482, 9, 669, 10, 746, 11, 654, 2, 7, 8, 12]
453$\\alpha^{2}$$\\alpha^{7}$$\\alpha^{8}$$\\alpha^{12}$38[0, 1, 3, 8, 4, 13, 5, 50, 6, 162, 7, 308, 8, 485, 9, 670, 10, 732, 11, 640, 2, 7, 8, 13]
454$\\alpha^{2}$$\\alpha^{7}$$\\alpha^{8}$$\\alpha^{13}$36[0, 1, 3, 6, 4, 12, 5, 62, 6, 176, 7, 290, 8, 446, 9, 666, 10, 768, 11, 666, 2, 7, 8, 14]
455$\\alpha^{2}$$\\alpha^{7}$$\\alpha^{8}$$\\alpha^{14}$34[0, 1, 3, 4, 4, 16, 5, 64, 6, 154, 7, 296, 8, 494, 9, 660, 10, 718, 11, 660, 2, 7, 9, 10]
456$\\alpha^{2}$$\\alpha^{7}$$\\alpha^{9}$$\\alpha^{10}$36[0, 1, 3, 6, 4, 11, 5, 55, 6, 172, 7, 310, 8, 465, 9, 655, 10, 752, 11, 650, 2, 7, 9, 11]
457$\\alpha^{2}$$\\alpha^{7}$$\\alpha^{9}$$\\alpha^{11}$39[0, 1, 3, 9, 4, 15, 5, 43, 6, 154, 7, 325, 8, 495, 9, 655, 10, 732, 11, 635, 2, 7, 9, 12]
458$\\alpha^{2}$$\\alpha^{7}$$\\alpha^{9}$$\\alpha^{12}$38[0, 1, 3, 8, 4, 14, 5, 49, 6, 158, 7, 312, 8, 490, 9, 665, 10, 732, 11, 640, 2, 7, 9, 13]
459$\\alpha^{2}$$\\alpha^{7}$$\\alpha^{9}$$\\alpha^{13}$35[0, 1, 3, 5, 4, 12, 5, 58, 6, 166, 7, 309, 8, 480, 9, 650, 10, 732, 11, 655, 2, 7, 9, 14]
460$\\alpha^{2}$$\\alpha^{7}$$\\alpha^{9}$$\\alpha^{14}$34[0, 1, 3, 4, 4, 10, 5, 63, 6, 172, 7, 298, 8, 480, 9, 665, 10, 712, 11, 640, 2, 7, 10, 11]
461$\\alpha^{2}$$\\alpha^{7}$$\\alpha^{10}$$\\alpha^{11}$31[0, 1, 3, 1, 4, 17, 5, 66, 6, 153, 7, 309, 8, 493, 9, 646, 10, 711, 11, 651, 2, 7, 10, 12]
462$\\alpha^{2}$$\\alpha^{7}$$\\alpha^{10}$$\\alpha^{12}$34[0, 1, 3, 4, 4, 11, 5, 61, 6, 173, 7, 304, 8, 467, 9, 657, 10, 735, 11, 652, 2, 7, 10, 13]
463$\\alpha^{2}$$\\alpha^{7}$$\\alpha^{10}$$\\alpha^{13}$31[0, 1, 3, 1, 4, 16, 5, 67, 6, 158, 7, 311, 8, 476, 9, 629, 10, 748, 11, 679, 2, 7, 10, 14]
464$\\alpha^{2}$$\\alpha^{7}$$\\alpha^{10}$$\\alpha^{14}$31[0, 1, 3, 1, 4, 13, 5, 71, 6, 168, 7, 293, 8, 466, 9, 659, 10, 752, 11, 659, 2, 7, 11, 12]
465$\\alpha^{2}$$\\alpha^{7}$$\\alpha^{11}$$\\alpha^{12}$311[0, 1, 3, 11, 4, 20, 5, 44, 6, 142, 7, 307, 8, 496, 9, 676, 10, 748, 11, 641, 2, 7, 11, 13]
466$\\alpha^{2}$$\\alpha^{7}$$\\alpha^{11}$$\\alpha^{13}$32[0, 1, 3, 2, 4, 20, 5, 53, 6, 164, 7, 312, 8, 466, 9, 687, 10, 712, 11, 618, 2, 7, 11, 14]
467$\\alpha^{2}$$\\alpha^{7}$$\\alpha^{11}$$\\alpha^{14}$31[0, 1, 3, 1, 4, 19, 5, 64, 6, 148, 7, 317, 8, 489, 9, 636, 10, 736, 11, 651, 2, 7, 12, 13]
468$\\alpha^{2}$$\\alpha^{7}$$\\alpha^{12}$$\\alpha^{13}$34[0, 1, 3, 4, 4, 11, 5, 63, 6, 170, 7, 298, 8, 475, 9, 665, 10, 732, 11, 640, 2, 7, 12, 14]
469$\\alpha^{2}$$\\alpha^{7}$$\\alpha^{12}$$\\alpha^{14}$34[0, 1, 3, 4, 4, 11, 5, 62, 6, 167, 7, 304, 8, 489, 9, 650, 10, 707, 11, 660, 2, 7, 13, 14]
470$\\alpha^{2}$$\\alpha^{7}$$\\alpha^{13}$$\\alpha^{14}$34[0, 1, 3, 4, 4, 11, 5, 62, 6, 170, 7, 304, 8, 475, 9, 650, 10, 732, 11, 660, 2, 8, 9, 10]
471$\\alpha^{2}$$\\alpha^{8}$$\\alpha^{9}$$\\alpha^{10}$36[0, 1, 3, 6, 4, 12, 5, 57, 6, 166, 7, 302, 8, 480, 9, 665, 10, 732, 11, 650, 2, 8, 9, 11]
472$\\alpha^{2}$$\\alpha^{8}$$\\alpha^{9}$$\\alpha^{11}$310[0, 1, 3, 10, 4, 16, 5, 45, 6, 154, 7, 310, 8, 488, 9, 673, 10, 740, 11, 638, 2, 8, 9, 12]
473$\\alpha^{2}$$\\alpha^{8}$$\\alpha^{9}$$\\alpha^{12}$38[0, 1, 3, 8, 4, 13, 5, 46, 6, 162, 7, 324, 8, 485, 9, 650, 10, 732, 11, 640, 2, 8, 9, 13]
474$\\alpha^{2}$$\\alpha^{8}$$\\alpha^{9}$$\\alpha^{13}$35[0, 1, 3, 5, 4, 14, 5, 56, 6, 160, 7, 313, 8, 486, 9, 652, 10, 730, 11, 647, 2, 8, 9, 14]
475$\\alpha^{2}$$\\alpha^{8}$$\\alpha^{9}$$\\alpha^{14}$36[0, 1, 3, 6, 4, 11, 5, 55, 6, 168, 7, 310, 8, 485, 9, 655, 10, 712, 11, 650, 2, 8, 10, 11]
476$\\alpha^{2}$$\\alpha^{8}$$\\alpha^{10}$$\\alpha^{11}$36[0, 1, 3, 6, 4, 13, 5, 56, 6, 159, 7, 306, 8, 499, 9, 660, 10, 707, 11, 650, 2, 8, 10, 12]
477$\\alpha^{2}$$\\alpha^{8}$$\\alpha^{10}$$\\alpha^{12}$34[0, 1, 3, 4, 4, 13, 5, 59, 6, 162, 7, 312, 8, 491, 9, 647, 10, 710, 11, 652, 2, 8, 10, 13]
478$\\alpha^{2}$$\\alpha^{8}$$\\alpha^{10}$$\\alpha^{13}$35[0, 1, 3, 5, 4, 12, 5, 59, 6, 165, 7, 305, 8, 484, 9, 655, 10, 727, 11, 655, 2, 8, 10, 14]
479$\\alpha^{2}$$\\alpha^{8}$$\\alpha^{10}$$\\alpha^{14}$34[0, 1, 3, 4, 4, 12, 5, 62, 6, 162, 7, 304, 8, 498, 9, 650, 10, 702, 11, 660, 2, 8, 11, 12]
480$\\alpha^{2}$$\\alpha^{8}$$\\alpha^{11}$$\\alpha^{12}$312[0, 1, 3, 12, 4, 20, 5, 41, 6, 142, 7, 308, 8, 496, 9, 681, 10, 748, 11, 636, 2, 8, 11, 13]
481$\\alpha^{2}$$\\alpha^{8}$$\\alpha^{11}$$\\alpha^{13}$36[0, 1, 3, 6, 4, 16, 5, 58, 6, 158, 7, 308, 8, 476, 9, 636, 10, 748, 11, 686, 2, 8, 11, 14]
482$\\alpha^{2}$$\\alpha^{8}$$\\alpha^{11}$$\\alpha^{14}$36[0, 1, 3, 6, 4, 17, 5, 57, 6, 151, 7, 300, 8, 495, 9, 675, 10, 723, 11, 630, 2, 8, 12, 13]
483$\\alpha^{2}$$\\alpha^{8}$$\\alpha^{12}$$\\alpha^{13}$34[0, 1, 3, 4, 4, 13, 5, 62, 6, 158, 7, 304, 8, 505, 9, 650, 10, 692, 11, 660, 2, 8, 12, 14]
484$\\alpha^{2}$$\\alpha^{8}$$\\alpha^{12}$$\\alpha^{14}$35[0, 1, 3, 5, 4, 15, 5, 57, 6, 155, 7, 309, 8, 497, 9, 657, 10, 715, 11, 647, 2, 8, 13, 14]
485$\\alpha^{2}$$\\alpha^{8}$$\\alpha^{13}$$\\alpha^{14}$38[0, 1, 3, 8, 4, 13, 5, 52, 6, 162, 7, 300, 8, 485, 9, 680, 10, 732, 11, 640, 2, 9, 10, 11]
486$\\alpha^{2}$$\\alpha^{9}$$\\alpha^{10}$$\\alpha^{11}$37[0, 1, 3, 7, 4, 14, 5, 55, 6, 162, 7, 305, 8, 478, 9, 653, 10, 740, 11, 673, 2, 9, 10, 12]
487$\\alpha^{2}$$\\alpha^{9}$$\\alpha^{10}$$\\alpha^{12}$32[0, 1, 3, 2, 4, 18, 5, 57, 6, 165, 7, 312, 8, 458, 9, 675, 10, 743, 11, 618, 2, 9, 10, 13]
488$\\alpha^{2}$$\\alpha^{9}$$\\alpha^{10}$$\\alpha^{13}$38[0, 1, 3, 8, 4, 15, 5, 55, 6, 161, 7, 296, 8, 475, 9, 671, 10, 743, 11, 656, 2, 9, 10, 14]
489$\\alpha^{2}$$\\alpha^{9}$$\\alpha^{10}$$\\alpha^{14}$32[0, 1, 3, 2, 4, 18, 5, 61, 6, 161, 7, 300, 8, 470, 9, 683, 10, 735, 11, 626, 2, 9, 11, 12]
490$\\alpha^{2}$$\\alpha^{9}$$\\alpha^{11}$$\\alpha^{12}$311[0, 1, 3, 11, 4, 20, 5, 48, 6, 146, 7, 295, 8, 484, 9, 684, 10, 756, 11, 649, 2, 9, 11, 13]
491$\\alpha^{2}$$\\alpha^{9}$$\\alpha^{11}$$\\alpha^{13}$36[0, 1, 3, 6, 4, 14, 5, 50, 6, 158, 7, 326, 8, 496, 9, 642, 10, 710, 11, 642, 2, 9, 11, 14]
492$\\alpha^{2}$$\\alpha^{9}$$\\alpha^{11}$$\\alpha^{14}$35[0, 1, 3, 5, 4, 14, 5, 63, 6, 157, 7, 293, 8, 502, 9, 663, 10, 695, 11, 663, 2, 9, 12, 13]
493$\\alpha^{2}$$\\alpha^{9}$$\\alpha^{12}$$\\alpha^{13}$31[0, 1, 3, 1, 4, 16, 5, 69, 6, 152, 7, 305, 8, 498, 9, 637, 10, 720, 11, 667, 2, 9, 12, 14]
494$\\alpha^{2}$$\\alpha^{9}$$\\alpha^{12}$$\\alpha^{14}$423[0, 1, 4, 23, 5, 56, 6, 154, 7, 320, 8, 479, 9, 664, 10, 700, 11, 640, 12, 537, 2, 9, 13, 14]
495$\\alpha^{2}$$\\alpha^{9}$$\\alpha^{13}$$\\alpha^{14}$35[0, 1, 3, 5, 4, 13, 5, 60, 6, 160, 7, 301, 8, 495, 9, 660, 10, 712, 11, 655, 2, 10, 11, 12]
496$\\alpha^{2}$$\\alpha^{10}$$\\alpha^{11}$$\\alpha^{12}$37[0, 1, 3, 7, 4, 14, 5, 56, 6, 162, 7, 299, 8, 478, 9, 668, 10, 740, 11, 653, 2, 10, 11, 13]
497$\\alpha^{2}$$\\alpha^{10}$$\\alpha^{11}$$\\alpha^{13}$32[0, 1, 3, 2, 4, 17, 5, 64, 6, 152, 7, 312, 8, 491, 9, 634, 10, 728, 11, 674, 2, 10, 11, 14]
498$\\alpha^{2}$$\\alpha^{10}$$\\alpha^{11}$$\\alpha^{14}$35[0, 1, 3, 5, 4, 11, 5, 61, 6, 173, 7, 303, 8, 467, 9, 643, 10, 735, 11, 683, 2, 10, 12, 13]
499$\\alpha^{2}$$\\alpha^{10}$$\\alpha^{12}$$\\alpha^{13}$31[0, 1, 3, 1, 4, 13, 5, 71, 6, 168, 7, 293, 8, 466, 9, 659, 10, 752, 11, 659, 2, 10, 12, 14]
" ], "text/plain": [ "" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "c_max = 500\n", "df = pd.DataFrame({'$\\\\alpha_2$': alpha_all[alpha_2[:c_max]], '$\\\\alpha_3$': alpha_all[alpha_3[:c_max]], '$\\\\alpha_4$': alpha_all[alpha_4[:c_max]], \n", " '$\\\\alpha_5$': alpha_all[alpha_5[:c_max]], '$d_{\\mathcal{D}}^\\perp$': d_all[:c_max], \n", " '$B_{d_{\\mathcal{D}}^\\perp}$': B_all[:c_max], 'Weight Enumerators* (the first 24 elements)': d_dual_b[:c_max]})\n", "\n", "pd.set_option('display.max_colwidth', 1000)\n", "pd.set_option('display.width', 800)\n", "(df.style\n", " .background_gradient(cmap=cm_1, subset=['$d_{\\mathcal{D}}^\\perp$','$B_{d_{\\mathcal{D}}^\\perp}$' ])\n", " .background_gradient(cmap=cm_2, subset=['$B_{d_{\\mathcal{D}}^\\perp}$' ])\n", " .set_caption('Tab. I All linear codes for (5,2)-SSS-based masking with $n=5$ shares over $\\mathbb{F}_{2^4}$.')\n", " .set_table_styles(styles))\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 2. Optimal linear codes for (5,2)-SSS-based masking" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 2.1 Linear codes with $d_{\\mathcal{D}}^\\perp=4$\n", "\n", "We focus on the the linear codes with greater $d_{\\mathcal{D}}^\\perp$, which are better in the sense of side-channel resistance (from our paper)." ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "ExecuteTime": { "end_time": "2020-10-18T20:19:58.543611Z", "start_time": "2020-10-18T20:19:58.534336Z" } }, "outputs": [], "source": [ "# Finding the indices of d_C=4\n", "d_index = []\n", "d_index_alpha_2 = []\n", "d_index_alpha_3 = []\n", "d_index_alpha_4 = []\n", "d_index_alpha_5 = []\n", "d_D_perp = 4\n", "for i in range(len(d_dual_b)):\n", " if d_dual_b[i][2] == d_D_perp:\n", " d_index.append(i)\n", " d_index_alpha_2.append(pow_ind[i][0])\n", " d_index_alpha_3.append(pow_ind[i][1])\n", " d_index_alpha_4.append(pow_ind[i][2])\n", " d_index_alpha_5.append(pow_ind[i][3])\n", "\n", "d_index = np.array(d_index)\n", "d_index_alpha_2 = np.array(d_index_alpha_2)\n", "d_index_alpha_3 = np.array(d_index_alpha_3)\n", "d_index_alpha_4 = np.array(d_index_alpha_4)\n", "d_index_alpha_5 = np.array(d_index_alpha_5)" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "ExecuteTime": { "end_time": "2020-10-18T20:20:04.970037Z", "start_time": "2020-10-18T20:20:04.961052Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "51\n", "[ 6 9 55 57 64 78 113 116 181 233 242 243 265 272 284 286 302 303\n", " 328 342 364 386 391 419 437 494 500 513 541 543 552 556 566 574 594 597\n", " 601 602 662 740 748 791 892 917 919 937 959 965 979 981 999]\n" ] } ], "source": [ "print(len(d_index))\n", "print(d_index)" ] }, { "cell_type": "code", "execution_count": 20, "metadata": { "ExecuteTime": { "end_time": "2020-10-18T20:23:57.188003Z", "start_time": "2020-10-18T20:23:57.181618Z" } }, "outputs": [], "source": [ "def highlight(s, threshold, column):\n", " is_min = pd.Series(data=False, index=s.index)\n", " is_min[column] = (s.loc[column] <= threshold)\n", " return ['background-color: gold' if is_min.any() else '' for v in is_min]" ] }, { "cell_type": "code", "execution_count": 21, "metadata": { "ExecuteTime": { "end_time": "2020-10-18T20:24:00.058078Z", "start_time": "2020-10-18T20:23:59.394117Z" }, "scrolled": false }, "outputs": [ { "data": { "text/html": [ "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " 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Tab. II Linear codes for (5,2)-SSS-based masking with $d_{\\mathcal{D}}^\\perp=4$.
$\\alpha_2$ $\\alpha_3$ $\\alpha_4$ $\\alpha_5$ $d_{\\mathcal{D}}^\\perp$ $B_{d_{\\mathcal{D}}^\\perp}$ Weight Enumerators* (the first 24 elements)
38$\\alpha^{3}$$\\alpha^{10}$$\\alpha^{11}$$\\alpha^{13}$417[0, 1, 4, 17, 5, 68, 6, 164, 7, 294, 8, 471, 9, 678, 10, 720, 11, 636, 12, 515, 3, 10, 11, 14]
27$\\alpha^{3}$$\\alpha^{4}$$\\alpha^{5}$$\\alpha^{13}$417[0, 1, 4, 17, 5, 70, 6, 151, 7, 308, 8, 495, 9, 634, 10, 723, 11, 664, 12, 491, 3, 4, 5, 14]
20$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{7}$$\\alpha^{14}$417[0, 1, 4, 17, 5, 68, 6, 164, 7, 294, 8, 471, 9, 678, 10, 720, 11, 636, 12, 515, 2, 4, 8, 9]
3$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{10}$$\\alpha^{12}$417[0, 1, 4, 17, 5, 70, 6, 151, 7, 308, 8, 495, 9, 634, 10, 723, 11, 664, 12, 491, 1, 2, 10, 13]
13$\\alpha^{1}$$\\alpha^{9}$$\\alpha^{11}$$\\alpha^{14}$417[0, 1, 4, 17, 5, 70, 6, 151, 7, 308, 8, 495, 9, 634, 10, 723, 11, 664, 12, 491, 1, 9, 12, 13]
5$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{5}$$\\alpha^{8}$417[0, 1, 4, 17, 5, 68, 6, 164, 7, 294, 8, 471, 9, 678, 10, 720, 11, 636, 12, 515, 1, 3, 5, 9]
45$\\alpha^{7}$$\\alpha^{8}$$\\alpha^{10}$$\\alpha^{12}$417[0, 1, 4, 17, 5, 68, 6, 164, 7, 294, 8, 471, 9, 678, 10, 720, 11, 636, 12, 515, 7, 8, 10, 13]
49$\\alpha^{8}$$\\alpha^{10}$$\\alpha^{13}$$\\alpha^{14}$417[0, 1, 4, 17, 5, 70, 6, 151, 7, 308, 8, 495, 9, 634, 10, 723, 11, 664, 12, 491, 8, 11, 12, 13]
23$\\alpha^{2}$$\\alpha^{5}$$\\alpha^{12}$$\\alpha^{13}$417[0, 1, 4, 17, 5, 68, 6, 164, 7, 294, 8, 471, 9, 678, 10, 720, 11, 636, 12, 515, 2, 5, 12, 14]
21$\\alpha^{2}$$\\alpha^{5}$$\\alpha^{6}$$\\alpha^{7}$417[0, 1, 4, 17, 5, 70, 6, 151, 7, 308, 8, 495, 9, 634, 10, 723, 11, 664, 12, 491, 2, 5, 6, 8]
12$\\alpha^{1}$$\\alpha^{8}$$\\alpha^{13}$$\\alpha^{14}$418[0, 1, 4, 18, 5, 72, 6, 145, 7, 300, 8, 510, 9, 644, 10, 703, 11, 664, 12, 506, 1, 9, 10, 11]
0$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{10}$418[0, 1, 4, 18, 5, 72, 6, 145, 7, 300, 8, 510, 9, 644, 10, 703, 11, 664, 12, 506, 1, 2, 3, 11]
41$\\alpha^{5}$$\\alpha^{6}$$\\alpha^{7}$$\\alpha^{8}$418[0, 1, 4, 18, 5, 72, 6, 145, 7, 300, 8, 510, 9, 644, 10, 703, 11, 664, 12, 506, 5, 6, 7, 9]
47$\\alpha^{7}$$\\alpha^{12}$$\\alpha^{13}$$\\alpha^{14}$418[0, 1, 4, 18, 5, 72, 6, 145, 7, 300, 8, 510, 9, 644, 10, 703, 11, 664, 12, 506, 8, 9, 10, 11]
2$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{9}$$\\alpha^{14}$418[0, 1, 4, 18, 5, 72, 6, 145, 7, 300, 8, 510, 9, 644, 10, 703, 11, 664, 12, 506, 1, 2, 10, 11]
22$\\alpha^{2}$$\\alpha^{5}$$\\alpha^{6}$$\\alpha^{12}$419[0, 1, 4, 19, 5, 68, 6, 148, 7, 308, 8, 492, 9, 648, 10, 728, 11, 648, 12, 492, 2, 5, 6, 13]
32$\\alpha^{3}$$\\alpha^{5}$$\\alpha^{8}$$\\alpha^{9}$419[0, 1, 4, 19, 5, 68, 6, 148, 7, 308, 8, 492, 9, 648, 10, 728, 11, 648, 12, 492, 3, 5, 8, 10]
42$\\alpha^{6}$$\\alpha^{7}$$\\alpha^{11}$$\\alpha^{14}$419[0, 1, 4, 19, 5, 71, 6, 141, 7, 304, 8, 515, 9, 639, 10, 703, 11, 664, 12, 501, 6, 7, 12, 13]
39$\\alpha^{4}$$\\alpha^{7}$$\\alpha^{8}$$\\alpha^{14}$419[0, 1, 4, 19, 5, 71, 6, 141, 7, 304, 8, 515, 9, 639, 10, 703, 11, 664, 12, 501, 4, 7, 9, 10]
28$\\alpha^{3}$$\\alpha^{4}$$\\alpha^{10}$$\\alpha^{11}$419[0, 1, 4, 19, 5, 71, 6, 141, 7, 304, 8, 515, 9, 639, 10, 703, 11, 664, 12, 501, 3, 4, 10, 12]
43$\\alpha^{6}$$\\alpha^{9}$$\\alpha^{11}$$\\alpha^{14}$419[0, 1, 4, 19, 5, 68, 6, 148, 7, 308, 8, 492, 9, 648, 10, 728, 11, 648, 12, 492, 6, 9, 12, 13]
10$\\alpha^{1}$$\\alpha^{7}$$\\alpha^{10}$$\\alpha^{12}$419[0, 1, 4, 19, 5, 68, 6, 148, 7, 308, 8, 492, 9, 648, 10, 728, 11, 648, 12, 492, 1, 7, 10, 13]
9$\\alpha^{1}$$\\alpha^{7}$$\\alpha^{8}$$\\alpha^{12}$419[0, 1, 4, 19, 5, 71, 6, 141, 7, 304, 8, 515, 9, 639, 10, 703, 11, 664, 12, 501, 1, 7, 8, 13]
8$\\alpha^{1}$$\\alpha^{5}$$\\alpha^{8}$$\\alpha^{9}$419[0, 1, 4, 19, 5, 71, 6, 141, 7, 304, 8, 515, 9, 639, 10, 703, 11, 664, 12, 501, 1, 5, 8, 10]
29$\\alpha^{3}$$\\alpha^{4}$$\\alpha^{10}$$\\alpha^{13}$419[0, 1, 4, 19, 5, 68, 6, 148, 7, 308, 8, 492, 9, 648, 10, 728, 11, 648, 12, 492, 3, 4, 10, 14]
19$\\alpha^{2}$$\\alpha^{4}$$\\alpha^{5}$$\\alpha^{7}$420[0, 1, 4, 20, 5, 64, 6, 150, 7, 326, 8, 472, 9, 618, 10, 764, 11, 668, 12, 472, 2, 4, 5, 8]
7$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{11}$$\\alpha^{13}$420[0, 1, 4, 20, 5, 64, 6, 150, 7, 326, 8, 472, 9, 618, 10, 764, 11, 668, 12, 472, 1, 3, 11, 14]
26$\\alpha^{2}$$\\alpha^{10}$$\\alpha^{12}$$\\alpha^{14}$420[0, 1, 4, 20, 5, 64, 6, 150, 7, 326, 8, 472, 9, 618, 10, 764, 11, 668, 12, 472, 2, 10, 13, 14]
48$\\alpha^{8}$$\\alpha^{10}$$\\alpha^{12}$$\\alpha^{13}$420[0, 1, 4, 20, 5, 64, 6, 150, 7, 326, 8, 472, 9, 618, 10, 764, 11, 668, 12, 472, 8, 10, 12, 14]
17$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{5}$$\\alpha^{13}$420[0, 1, 4, 20, 5, 64, 6, 150, 7, 326, 8, 472, 9, 618, 10, 764, 11, 668, 12, 472, 2, 3, 5, 14]
50$\\alpha^{10}$$\\alpha^{12}$$\\alpha^{13}$$\\alpha^{14}$421[0, 1, 4, 21, 5, 63, 6, 152, 7, 312, 8, 479, 9, 655, 10, 728, 11, 648, 12, 503, 11, 12, 13, 14]
1$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{13}$421[0, 1, 4, 21, 5, 63, 6, 152, 7, 312, 8, 479, 9, 655, 10, 728, 11, 648, 12, 503, 1, 2, 3, 14]
4$\\alpha^{1}$$\\alpha^{2}$$\\alpha^{12}$$\\alpha^{14}$421[0, 1, 4, 21, 5, 63, 6, 152, 7, 312, 8, 479, 9, 655, 10, 728, 11, 648, 12, 503, 1, 2, 13, 14]
14$\\alpha^{1}$$\\alpha^{11}$$\\alpha^{13}$$\\alpha^{14}$421[0, 1, 4, 21, 5, 63, 6, 152, 7, 312, 8, 479, 9, 655, 10, 728, 11, 648, 12, 503, 1, 12, 13, 14]
15$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{4}$$\\alpha^{5}$421[0, 1, 4, 21, 5, 63, 6, 152, 7, 312, 8, 479, 9, 655, 10, 728, 11, 648, 12, 503, 2, 3, 4, 6]
40$\\alpha^{4}$$\\alpha^{7}$$\\alpha^{10}$$\\alpha^{13}$423[0, 1, 4, 23, 5, 61, 6, 145, 7, 316, 8, 491, 9, 657, 10, 711, 11, 640, 12, 521, 4, 7, 10, 14]
46$\\alpha^{7}$$\\alpha^{10}$$\\alpha^{12}$$\\alpha^{13}$423[0, 1, 4, 23, 5, 56, 6, 154, 7, 320, 8, 479, 9, 664, 10, 700, 11, 640, 12, 537, 7, 10, 12, 14]
6$\\alpha^{1}$$\\alpha^{3}$$\\alpha^{10}$$\\alpha^{13}$423[0, 1, 4, 23, 5, 56, 6, 154, 7, 320, 8, 479, 9, 664, 10, 700, 11, 640, 12, 537, 1, 3, 10, 14]
30$\\alpha^{3}$$\\alpha^{5}$$\\alpha^{6}$$\\alpha^{8}$423[0, 1, 4, 23, 5, 56, 6, 154, 7, 320, 8, 479, 9, 664, 10, 700, 11, 640, 12, 537, 3, 5, 6, 9]
25$\\alpha^{2}$$\\alpha^{9}$$\\alpha^{12}$$\\alpha^{14}$423[0, 1, 4, 23, 5, 56, 6, 154, 7, 320, 8, 479, 9, 664, 10, 700, 11, 640, 12, 537, 2, 9, 13, 14]
36$\\alpha^{3}$$\\alpha^{6}$$\\alpha^{9}$$\\alpha^{11}$423[0, 1, 4, 23, 5, 61, 6, 145, 7, 316, 8, 491, 9, 657, 10, 711, 11, 640, 12, 521, 3, 6, 9, 12]
35$\\alpha^{3}$$\\alpha^{6}$$\\alpha^{8}$$\\alpha^{12}$423[0, 1, 4, 23, 5, 61, 6, 145, 7, 316, 8, 491, 9, 657, 10, 711, 11, 640, 12, 521, 3, 6, 8, 13]
33$\\alpha^{3}$$\\alpha^{5}$$\\alpha^{9}$$\\alpha^{12}$423[0, 1, 4, 23, 5, 61, 6, 145, 7, 316, 8, 491, 9, 657, 10, 711, 11, 640, 12, 521, 3, 5, 9, 13]
24$\\alpha^{2}$$\\alpha^{6}$$\\alpha^{9}$$\\alpha^{12}$423[0, 1, 4, 23, 5, 61, 6, 145, 7, 316, 8, 491, 9, 657, 10, 711, 11, 640, 12, 521, 2, 6, 9, 13]
16$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{5}$$\\alpha^{12}$423[0, 1, 4, 23, 5, 56, 6, 154, 7, 320, 8, 479, 9, 664, 10, 700, 11, 640, 12, 537, 2, 3, 5, 13]
18$\\alpha^{2}$$\\alpha^{3}$$\\alpha^{9}$$\\alpha^{12}$424[0, 1, 4, 24, 5, 60, 6, 134, 7, 336, 8, 498, 9, 620, 10, 742, 11, 640, 12, 480, 2, 3, 9, 13]
11$\\alpha^{1}$$\\alpha^{7}$$\\alpha^{10}$$\\alpha^{13}$424[0, 1, 4, 24, 5, 60, 6, 134, 7, 336, 8, 498, 9, 620, 10, 742, 11, 640, 12, 480, 1, 7, 10, 14]
44$\\alpha^{6}$$\\alpha^{9}$$\\alpha^{12}$$\\alpha^{14}$424[0, 1, 4, 24, 5, 60, 6, 134, 7, 336, 8, 498, 9, 620, 10, 742, 11, 640, 12, 480, 6, 9, 13, 14]
34$\\alpha^{3}$$\\alpha^{6}$$\\alpha^{8}$$\\alpha^{9}$424[0, 1, 4, 24, 5, 60, 6, 134, 7, 336, 8, 498, 9, 620, 10, 742, 11, 640, 12, 480, 3, 6, 8, 10]
31$\\alpha^{3}$$\\alpha^{5}$$\\alpha^{6}$$\\alpha^{12}$424[0, 1, 4, 24, 5, 60, 6, 134, 7, 336, 8, 498, 9, 620, 10, 742, 11, 640, 12, 480, 3, 5, 6, 13]
37$\\alpha^{3}$$\\alpha^{6}$$\\alpha^{9}$$\\alpha^{12}$425[0, 1, 4, 25, 5, 64, 6, 120, 7, 320, 8, 550, 9, 640, 10, 656, 11, 640, 12, 550, 3, 6, 9, 13]
" ], "text/plain": [ "" ] }, "execution_count": 21, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df_4 = pd.DataFrame({'$\\\\alpha_2$': np.array(alpha_all)[d_index_alpha_2], '$\\\\alpha_3$': np.array(alpha_all)[d_index_alpha_3], \n", " '$\\\\alpha_4$': np.array(alpha_all)[d_index_alpha_4], '$\\\\alpha_5$': np.array(alpha_all)[d_index_alpha_5],\n", " '$d_{\\mathcal{D}}^\\perp$': d_all[d_index], '$B_{d_{\\mathcal{D}}^\\perp}$': B_all[d_index], 'Weight Enumerators* (the first 24 elements)': np.array(d_dual_b)[d_index]})\n", "df_4 = df_4.sort_values(by=['$B_{d_{\\mathcal{D}}^\\perp}$'], ascending=True)\n", "\n", "(df_4.style\n", " .apply(highlight, threshold=17, column=['$B_{d_{\\mathcal{D}}^\\perp}$'], axis=1)\n", " .background_gradient(cmap=cm_2, subset=['$B_{d_{\\mathcal{D}}^\\perp}$' ])\n", " .set_caption('Tab. II Linear codes for (5,2)-SSS-based masking with $d_{\\mathcal{D}}^\\perp=4$.')\n", " .set_table_styles(styles))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 2.2 Optimal codes for (5,2)-SSS-based masking\n", "\n", "As shown in our paper, the codes satifying two conditions are optimal:\n", "\n", "- Maximizing $d_{\\mathcal{D}}^\\perp$, here $\\max\\{d_{\\mathcal{D}}^\\perp\\} = 4$\n", "- Minimizing $B_{d_{\\mathcal{D}}^\\perp}$, here $\\min\\{B_{d_{\\mathcal{D}}^\\perp}\\} = 17$\n", "\n", "Note that we use two complementary metrics **SNR** (signal-to-noise ratio) and **MI** (mutual information) to assess the side-channel resistance of SSS-based masking with different codes." ] }, { "cell_type": "markdown", "metadata": { "scrolled": false }, "source": [ "As a result of Tab. II, we conclude that the optimal codes for (5,2)-SSS based masking are generated by $\\mathbf{H} = \\left(\\begin{matrix} \\alpha_1 & \\alpha_2 & \\alpha_3 & \\alpha_4 & \\alpha_5 \\\\ \\alpha_1^2 & \\alpha_2^2 & \\alpha_3^2 & \\alpha_4^2 & \\alpha_5^2 \\end{matrix}\\right)$ where there are nine optimal codes with different $(\\alpha_1, \\alpha_2, \\alpha_3, \\alpha_4, \\alpha_5)$. Note that permutation on five public points does not change the codes due to equivalence.\n", "\n", "The generator matrices of the *all ten* codes are: " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "- $(\\alpha_1, \\alpha_3, \\alpha_3, \\alpha_4, \\alpha_5) = (1, \\alpha^3, \\alpha^{10}, \\alpha^{11}, \\alpha^{13})$\n", "$$\n", "\\mathbf{H}_{1}=\\left( \\begin{matrix} 1 & \\alpha^{3} & \\alpha^{10} & \\alpha^{11} & \\alpha^{13} \\\\ 1 & \\alpha^6 & \\alpha^{5} & \\alpha^{7} &\\alpha^{11} \\end{matrix} \\right) \\in \\mathbb{F}_{2^4}^{2\\times 5} \n", "= \\left(\n", " \\begin{matrix}\n", " 1~~0~~0~~0~~0~~0~~0~~0~~1~~0~~1~~0~~0~~0~~1~~0~~1~~1~~0~~1 \\\\\n", " 0~~1~~0~~0~~0~~0~~0~~0~~0~~1~~0~~1~~0~~0~~0~~1~~1~~0~~1~~0 \\\\\n", " 0~~0~~1~~0~~0~~0~~0~~0~~1~~1~~1~~0~~1~~1~~0~~0~~0~~1~~0~~1 \\\\\n", " 0~~0~~0~~1~~0~~0~~0~~0~~0~~1~~1~~1~~0~~1~~1~~0~~1~~1~~1~~0 \\\\\n", " 0~~0~~0~~0~~1~~0~~0~~0~~1~~0~~1~~1~~0~~0~~1~~1~~0~~0~~1~~0 \\\\\n", " 0~~0~~0~~0~~0~~1~~0~~0~~1~~0~~0~~1~~1~~1~~0~~1~~0~~0~~0~~1 \\\\\n", " 0~~0~~0~~0~~0~~0~~1~~0~~1~~0~~0~~0~~1~~0~~1~~0~~1~~1~~0~~0 \\\\\n", " 0~~0~~0~~0~~0~~0~~0~~1~~0~~1~~0~~0~~0~~1~~0~~1~~0~~1~~1~~0 \n", " \\end{matrix} \n", "\\right) \\normalsize\\in \\mathbb{F}_2^{8\\times 20}\n", "$$ " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "- $(\\alpha_1, \\alpha_2, \\alpha_3, \\alpha_4, \\alpha_5) = (1, \\alpha^3, \\alpha^4, \\alpha^{5}, \\alpha^{13})$\n", "$$\n", "\\mathbf{H}_{2}=\\left( \\begin{matrix} 1 & \\alpha^3 & \\alpha^4 & \\alpha^{5} & \\alpha^{13} \\\\ 1 & \\alpha^6 & \\alpha^{8} & \\alpha^{10} &\\alpha^{11} \\end{matrix} \\right) \\in \\mathbb{F}_{2^4}^{2\\times 5} \n", "= \\left(\n", " \\begin{matrix}\n", " 1~~0~~0~~0~~0~~0~~0~~0~~1~~1~~1~~1~~0~~0~~1~~0~~1~~1~~0~~1 \\\\\n", " 0~~1~~0~~0~~0~~0~~0~~0~~1~~0~~1~~1~~0~~0~~0~~1~~1~~0~~1~~0 \\\\\n", " 0~~0~~1~~0~~0~~0~~0~~0~~1~~0~~0~~1~~1~~1~~0~~0~~0~~1~~0~~1 \\\\\n", " 0~~0~~0~~1~~0~~0~~0~~0~~1~~0~~0~~0~~0~~1~~1~~0~~1~~1~~1~~0 \\\\\n", " 0~~0~~0~~0~~1~~0~~0~~0~~0~~0~~0~~1~~1~~0~~1~~1~~0~~0~~1~~0 \\\\\n", " 0~~0~~0~~0~~0~~1~~0~~0~~1~~1~~0~~0~~1~~0~~0~~1~~0~~0~~0~~1 \\\\\n", " 0~~0~~0~~0~~0~~0~~1~~0~~0~~1~~1~~0~~1~~0~~0~~0~~1~~1~~0~~0 \\\\\n", " 0~~0~~0~~0~~0~~0~~0~~1~~0~~0~~1~~1~~0~~1~~0~~0~~0~~1~~1~~0 \n", " \\end{matrix} \n", "\\right) \\normalsize\\in \\mathbb{F}_2^{8\\times 20}\n", "$$ " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "- $(\\alpha_1, \\alpha_2, \\alpha_3, \\alpha_4, \\alpha_5) = (1, \\alpha^2, \\alpha^4, \\alpha^{7}, \\alpha^{14})$\n", "$$\n", "\\mathbf{H}_{3}=\\left( \\begin{matrix} 1 & \\alpha^2 & \\alpha^4 & \\alpha^{7} & \\alpha^{14} \\\\ 1 & \\alpha^4 & \\alpha^{8} & \\alpha^{14} &\\alpha^{13} \\end{matrix} \\right) \\in \\mathbb{F}_{2^4}^{2\\times 5} \n", "= \\left(\n", " \\begin{matrix}\n", " 1~~0~~0~~0~~0~~0~~0~~0~~0~~0~~1~~1~~0~~1~~1~~1~~1~~1~~0~~0\\\\\n", " 0~~1~~0~~0~~0~~0~~0~~0~~1~~1~~0~~1~~1~~1~~1~~1~~0~~1~~1~~0\\\\\n", " 0~~0~~1~~0~~0~~0~~0~~0~~1~~0~~1~~0~~1~~0~~1~~1~~0~~0~~1~~1\\\\\n", " 0~~0~~0~~1~~0~~0~~0~~0~~0~~1~~0~~1~~1~~0~~0~~1~~1~~1~~0~~1\\\\\n", " 0~~0~~0~~0~~1~~0~~0~~0~~1~~1~~1~~0~~0~~0~~1~~1~~1~~1~~0~~1\\\\\n", " 0~~0~~0~~0~~0~~1~~0~~0~~0~~1~~1~~1~~1~~1~~0~~1~~1~~0~~1~~0\\\\\n", " 0~~0~~0~~0~~0~~0~~1~~0~~1~~1~~1~~1~~1~~0~~1~~0~~0~~1~~0~~1\\\\\n", " 0~~0~~0~~0~~0~~0~~0~~1~~1~~0~~1~~1~~0~~1~~0~~1~~1~~1~~1~~0 \n", " \\end{matrix} \n", "\\right) \\normalsize\\in \\mathbb{F}_2^{8\\times 20}\n", "$$ " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "- $(\\alpha_1, \\alpha_2, \\alpha_3, \\alpha_4, \\alpha_5) = (1, \\alpha^1, \\alpha^2, \\alpha^{10}, \\alpha^{12})$\n", "$$\n", "\\mathbf{H}_{4}=\\left( \\begin{matrix} 1 & \\alpha^1 & \\alpha^2 & \\alpha^{10} & \\alpha^{12} \\\\ 1 & \\alpha^2 & \\alpha^{4} & \\alpha^{5} & \\alpha^{9} \\end{matrix} \\right) \\in \\mathbb{F}_{2^4}^{2\\times 5} \n", "= \\left(\n", " \\begin{matrix}\n", " 1~~0~~0~~0~~0~~0~~0~~0~~0~~0~~0~~1~~1~~0~~0~~1~~0~~0~~1~~1 \\\\\n", " 0~~1~~0~~0~~0~~0~~0~~0~~1~~1~~0~~0~~1~~0~~0~~0~~1~~1~~0~~1 \\\\\n", " 0~~0~~1~~0~~0~~0~~0~~0~~0~~1~~1~~0~~0~~1~~0~~0~~1~~0~~1~~0 \\\\\n", " 0~~0~~0~~1~~0~~0~~0~~0~~0~~0~~1~~1~~0~~0~~1~~0~~0~~1~~0~~1 \\\\\n", " 0~~0~~0~~0~~1~~0~~0~~0~~0~~1~~1~~0~~1~~1~~1~~0~~0~~0~~0~~1 \\\\\n", " 0~~0~~0~~0~~0~~1~~0~~0~~0~~0~~1~~1~~0~~1~~1~~1~~1~~1~~0~~0 \\\\\n", " 0~~0~~0~~0~~0~~0~~1~~0~~1~~1~~0~~1~~1~~1~~1~~1~~0~~1~~1~~0 \\\\\n", " 0~~0~~0~~0~~0~~0~~0~~1~~1~~0~~1~~0~~1~~0~~1~~1~~0~~0~~1~~1 \n", " \\end{matrix} \n", "\\right) \\normalsize\\in \\mathbb{F}_2^{8\\times 20}\n", "$$ \n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "- $(\\alpha_1, \\alpha_2, \\alpha_3, \\alpha_4, \\alpha_5) = (1, \\alpha^1, \\alpha^9, \\alpha^{11}, \\alpha^{14})$\n", "$$\n", "\\mathbf{H}_{5}=\\left( \\begin{matrix} 1 & \\alpha^1 & \\alpha^9 & \\alpha^{11} & \\alpha^{14} \\\\ 1 & \\alpha^2 & \\alpha^{3} & \\alpha^{7} &\\alpha^{13} \\end{matrix} \\right) \\in \\mathbb{F}_{2^4}^{2\\times 5} \n", "= \\left(\n", " \\begin{matrix}\n", " 1~~0~~0~~0~~0~~0~~0~~0~~1~~0~~1~~0~~1~~0~~1~~1~~0~~0~~1~~0\\\\\n", " 0~~1~~0~~0~~0~~0~~0~~0~~0~~1~~0~~1~~1~~0~~0~~1~~0~~0~~0~~1\\\\\n", " 0~~0~~1~~0~~0~~0~~0~~0~~1~~1~~1~~0~~1~~0~~0~~0~~1~~1~~0~~0\\\\\n", " 0~~0~~0~~1~~0~~0~~0~~0~~0~~1~~1~~1~~0~~1~~0~~0~~0~~1~~1~~0\\\\\n", " 0~~0~~0~~0~~1~~0~~0~~0~~0~~1~~1~~1~~0~~0~~0~~1~~1~~1~~1~~1\\\\\n", " 0~~0~~0~~0~~0~~1~~0~~0~~1~~1~~1~~1~~1~~1~~0~~0~~1~~0~~1~~1\\\\\n", " 0~~0~~0~~0~~0~~0~~1~~0~~1~~0~~1~~1~~0~~1~~1~~0~~1~~0~~0~~1\\\\\n", " 0~~0~~0~~0~~0~~0~~0~~1~~1~~0~~0~~1~~0~~0~~1~~1~~1~~0~~0~~0 \n", " \\end{matrix} \n", "\\right) \\normalsize\\in \\mathbb{F}_2^{8\\times 20}\n", "$$ " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "- $(\\alpha_1, \\alpha_2, \\alpha_3, \\alpha_4, \\alpha_5) = (1, \\alpha^1, \\alpha^3, \\alpha^{5}, \\alpha^{8})$\n", "$$\n", "\\mathbf{H}_{6}=\\left( \\begin{matrix} 1 & \\alpha^1 & \\alpha^3 & \\alpha^{5} & \\alpha^{8} \\\\ 1 & \\alpha^{2} & \\alpha^{6} & \\alpha^{10} & \\alpha^{1} \\end{matrix} \\right) \\in \\mathbb{F}_{2^4}^{2\\times 5} \n", "= \\left(\n", " \\begin{matrix}\n", " 1~~0~~0~~0~~0~~0~~0~~0~~1~~0~~1~~0~~0~~0~~0~~1~~1~~0~~0~~1 \\\\\n", " 0~~1~~0~~0~~0~~0~~0~~0~~0~~1~~0~~1~~1~~1~~0~~0~~1~~0~~0~~0 \\\\\n", " 0~~0~~1~~0~~0~~0~~0~~0~~1~~1~~1~~0~~0~~1~~1~~0~~0~~1~~0~~0 \\\\\n", " 0~~0~~0~~1~~0~~0~~0~~0~~0~~1~~1~~1~~0~~0~~1~~1~~0~~0~~1~~0 \\\\\n", " 0~~0~~0~~0~~1~~0~~0~~0~~1~~1~~1~~1~~1~~1~~1~~0~~0~~1~~1~~0 \\\\\n", " 0~~0~~0~~0~~0~~1~~0~~0~~1~~0~~1~~1~~0~~1~~1~~1~~0~~0~~1~~1 \\\\\n", " 0~~0~~0~~0~~0~~0~~1~~0~~1~~0~~0~~1~~1~~1~~1~~1~~1~~1~~0~~1 \\\\\n", " 0~~0~~0~~0~~0~~0~~0~~1~~1~~0~~0~~0~~1~~0~~1~~1~~1~~0~~1~~0 \n", " \\end{matrix} \n", "\\right) \\normalsize\\in \\mathbb{F}_2^{8\\times 20}\n", "$$ \n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "- $(\\alpha_1, \\alpha_2, \\alpha_3, \\alpha_4, \\alpha_5) = (1, \\alpha^7, \\alpha^8, \\alpha^{10}, \\alpha^{12})$\n", "$$\n", "\\mathbf{H}_{7}=\\left( \\begin{matrix} 1 & \\alpha^7 & \\alpha^8 & \\alpha^{10} & \\alpha^{12} \\\\ 1 & \\alpha^{14} & \\alpha^{1} & \\alpha^{5} & \\alpha^{9} \\end{matrix} \\right) \\in \\mathbb{F}_{2^4}^{2\\times 5} \n", "= \\left(\n", " \\begin{matrix}\n", " 1~~0~~0~~0~~0~~0~~0~~0~~1~~1~~1~~0~~1~~1~~0~~1~~0~~1~~1~~0 \\\\\n", " 0~~1~~0~~0~~0~~0~~0~~0~~0~~1~~1~~1~~1~~0~~1~~0~~0~~0~~1~~1 \\\\\n", " 0~~0~~1~~0~~0~~0~~0~~0~~1~~1~~1~~1~~0~~1~~0~~1~~1~~1~~0~~1 \\\\\n", " 0~~0~~0~~1~~0~~0~~0~~0~~1~~0~~1~~1~~1~~1~~1~~0~~1~~0~~1~~0 \\\\\n", " 0~~0~~0~~0~~1~~0~~0~~0~~0~~1~~0~~1~~1~~0~~0~~1~~1~~1~~0~~1 \\\\\n", " 0~~0~~0~~0~~0~~1~~0~~0~~1~~1~~1~~0~~1~~0~~0~~0~~1~~0~~1~~0 \\\\\n", " 0~~0~~0~~0~~0~~0~~1~~0~~0~~1~~1~~1~~0~~1~~0~~0~~0~~1~~0~~1 \\\\\n", " 0~~0~~0~~0~~0~~0~~0~~1~~1~~1~~1~~1~~0~~0~~1~~0~~1~~1~~1~~0 \n", " \\end{matrix} \n", "\\right) \\normalsize\\in \\mathbb{F}_2^{8\\times 20}\n", "$$ \n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "- $(\\alpha_1, \\alpha_2, \\alpha_3, \\alpha_4, \\alpha_5) = (1, \\alpha^8, \\alpha^{10}, \\alpha^{13}, \\alpha^{14})$\n", "$$\n", "\\mathbf{H}_{8}=\\left( \\begin{matrix} 1 & \\alpha^8 & \\alpha^{10} & \\alpha^{13} & \\alpha^{14} \\\\ 1 & \\alpha^1 & \\alpha^{5} & \\alpha^{11} &\\alpha^{13} \\end{matrix} \\right) \\in \\mathbb{F}_{2^4}^{2\\times 5} \n", "= \\left(\n", " \\begin{matrix}\n", " 1~~0~~0~~0~~0~~0~~0~~0~~0~~1~~0~~1~~1~~0~~0~~1~~0~~0~~0~~1\\\\\n", " 0~~1~~0~~0~~0~~0~~0~~0~~1~~1~~1~~0~~1~~0~~0~~0~~1~~1~~0~~0\\\\\n", " 0~~0~~1~~0~~0~~0~~0~~0~~0~~1~~1~~1~~0~~1~~0~~0~~0~~1~~1~~0\\\\\n", " 0~~0~~0~~1~~0~~0~~0~~0~~1~~1~~1~~1~~0~~0~~1~~0~~0~~0~~1~~1\\\\\n", " 0~~0~~0~~0~~1~~0~~0~~0~~0~~1~~1~~0~~0~~1~~0~~1~~1~~1~~0~~1\\\\\n", " 0~~0~~0~~0~~0~~1~~0~~0~~0~~0~~1~~1~~1~~1~~1~~0~~1~~0~~1~~0\\\\\n", " 0~~0~~0~~0~~0~~0~~1~~0~~1~~1~~0~~1~~0~~1~~1~~1~~0~~1~~0~~1\\\\\n", " 0~~0~~0~~0~~0~~0~~0~~1~~1~~0~~1~~0~~1~~1~~1~~1~~1~~1~~1~~0 \n", " \\end{matrix} \n", "\\right) \\normalsize\\in \\mathbb{F}_2^{8\\times 20}\n", "$$ " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "- $(\\alpha_1, \\alpha_2, \\alpha_3, \\alpha_4, \\alpha_5) = (1, \\alpha^2, \\alpha^5, \\alpha^{12}, \\alpha^{13})$\n", "$$\n", "\\mathbf{H}_{9}=\\left( \\begin{matrix} 1 & \\alpha^2 & \\alpha^5 & \\alpha^{12} & \\alpha^{13} \\\\ 1 & \\alpha^4 & \\alpha^{10} & \\alpha^{9} &\\alpha^{11} \\end{matrix} \\right) \\in \\mathbb{F}_{2^4}^{2\\times 5} \n", "= \\left(\n", " \\begin{matrix}\n", " 1~~0~~0~~0~~0~~0~~0~~0~~1~~0~~1~~1~~0~~1~~1~~1~~1~~1~~0~~0 \\\\\n", " 0~~1~~0~~0~~0~~0~~0~~0~~1~~0~~0~~1~~1~~1~~1~~1~~0~~1~~1~~0 \\\\\n", " 0~~0~~1~~0~~0~~0~~0~~0~~1~~0~~0~~0~~1~~0~~1~~1~~0~~0~~1~~1 \\\\\n", " 0~~0~~0~~1~~0~~0~~0~~0~~0~~1~~0~~0~~1~~0~~0~~1~~1~~1~~0~~1 \\\\\n", " 0~~0~~0~~0~~1~~0~~0~~0~~0~~1~~1~~0~~1~~0~~1~~1~~0~~1~~0~~1 \\\\\n", " 0~~0~~0~~0~~0~~1~~0~~0~~0~~0~~1~~1~~1~~0~~0~~1~~1~~1~~1~~0 \\\\\n", " 0~~0~~0~~0~~0~~0~~1~~0~~1~~1~~0~~1~~1~~0~~0~~0~~0~~1~~1~~1 \\\\\n", " 0~~0~~0~~0~~0~~0~~0~~1~~1~~0~~1~~0~~0~~1~~0~~0~~1~~1~~1~~1 \n", " \\end{matrix} \n", "\\right) \\normalsize\\in \\mathbb{F}_2^{8\\times 20}\n", "$$ " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "- $(\\alpha_1, \\alpha_2, \\alpha_3, \\alpha_4, \\alpha_5) = (1, \\alpha^2, \\alpha^5, \\alpha^{6}, \\alpha^{7})$\n", "$$\n", "\\mathbf{H}_{10}=\\left( \\begin{matrix} 1 & \\alpha^2 & \\alpha^5 & \\alpha^{6} & \\alpha^{7} \\\\ 1 & \\alpha^4 & \\alpha^{10} & \\alpha^{12} &\\alpha^{14} \\end{matrix} \\right) \\in \\mathbb{F}_{2^4}^{2\\times 5} \n", "= \\left(\n", " \\begin{matrix}\n", " 1~~0~~0~~0~~0~~0~~0~~0~~1~~0~~1~~1~~0~~1~~0~~0~~0~~1~~1~~1\\\\\n", " 0~~1~~0~~0~~0~~0~~0~~0~~1~~0~~0~~1~~0~~0~~1~~0~~1~~1~~1~~1\\\\\n", " 0~~0~~1~~0~~0~~0~~0~~0~~1~~0~~0~~0~~0~~0~~0~~1~~1~~0~~1~~1\\\\\n", " 0~~0~~0~~1~~0~~0~~0~~0~~0~~1~~0~~0~~1~~1~~0~~0~~1~~0~~0~~1\\\\\n", " 0~~0~~0~~0~~1~~0~~0~~0~~0~~1~~1~~0~~0~~1~~0~~1~~0~~0~~1~~1\\\\\n", " 0~~0~~0~~0~~0~~1~~0~~0~~0~~0~~1~~1~~1~~1~~1~~0~~1~~1~~0~~1\\\\\n", " 0~~0~~0~~0~~0~~0~~1~~0~~1~~1~~0~~1~~0~~1~~1~~1~~1~~0~~1~~0\\\\\n", " 0~~0~~0~~0~~0~~0~~0~~1~~1~~0~~1~~0~~1~~1~~1~~1~~0~~1~~0~~1 \n", " \\end{matrix} \n", "\\right) \\normalsize\\in \\mathbb{F}_2^{8\\times 20}\n", "$$ " ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.7.3" }, "toc": { "base_numbering": 1, "nav_menu": {}, "number_sections": false, "sideBar": true, "skip_h1_title": false, "title_cell": "Table of Contents", "title_sidebar": "Contents", "toc_cell": false, "toc_position": {}, "toc_section_display": true, "toc_window_display": false } }, "nbformat": 4, "nbformat_minor": 2 }