# American Institute of Mathematical Sciences

doi: 10.3934/amc.2022014
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## Self-orthogonal codes from equitable partitions of distance-regular graphs

 Department of Mathematics, University of Rijeka, Croatia

* Corresponding author: Dean Crnković

Received  November 2021 Early access March 2022

Fund Project: This work has been fully supported by Croatian Science Foundation under the project 6732

We give two methods for a construction of self-orthogonal linear codes from equitable partitions of distance-regular graphs. By applying these methods, we construct self-orthogonal codes from equitable partitions of the graph of unitals in $PG(2,4)$ and the only known strongly regular graph with parameters $(216,40,4,8)$. Some of the codes obtained are optimal.

Citation: Dean Crnković, Sanja Rukavina, Andrea Švob. Self-orthogonal codes from equitable partitions of distance-regular graphs. Advances in Mathematics of Communications, doi: 10.3934/amc.2022014
##### References:
 [1] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I, The user language, J. Symb. Comput., 24 (1997), 235-265.  doi: 10.1006/jsco.1996.0125. [2] A. E. Brouwer, A. M. Cohen and A. Neumaier, Distance-Regular Graphs, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-642-74341-2. [3] D. Crnković, M. Maksimović, B. G. Rodrigues and S. Rukavina, Self-orthogonal codes from the strongly regular graphs on up to 40 vertices, Adv. Math. Commun., 10 (2016), 555-582.  doi: 10.3934/amc.2016026. [4] D. Crnković, F. Pavese and A. Švob, On the PSU(4, 2)-invariant vertex-transitive strongly regular (216, 40, 4, 8) graph, Graphs Combin., 36 (2020), 503-513.  doi: 10.1007/s00373-020-02132-5. [5] D. Crnković, B. G. Rodrigues, S. Rukavina and L. Simčić, Self-orthogonal codes from orbit matrices of 2-designs, Adv. Math. Commun., 7 (2013), 161-174.  doi: 10.3934/amc.2013.7.161. [6] D. Crnković, S. Rukavina and A. Švob, Self-orthogonal codes from equitable partitions of association schemes, J. Algebraic Combin., 55 (2022), 157-171.  doi: 10.1007/s10801-021-01104-z. [7] D. Crnković, S. Rukavina and A. Švob, New strongly regular graphs from orthogonal groups $O^+(6, 2)$ and $O^-(6, 2)$, Discrete Math., 341 (2018), 2723-2728.  doi: 10.1016/j.disc.2018.06.029. [8] C. D. Godsil and W. J. Martin, Quotients of association schemes, J. Combin. Theory Ser. A, 69 (1995), 185-199.  doi: 10.1016/0097-3165(95)90050-0. [9] M. Grassl, Bounds on the minimum distance of linear codes and quantum codes, Accessed 24 October 2021, http://www.codetables.de. [10] A. Hanaki, Elementary functions for association schemes on GAP, 2013, http://math.shinshu-u.ac.jp/＄\sim＄hanaki/as/gap/association_scheme.pdf [11] M. Harada and V. Tonchev, Self-orthogonal codes from symmetric designs with fixed-point-free automorphisms, Discrete Math., 264 (2003), 81-90.  doi: 10.1016/S0012-365X(02)00553-8. [12] W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511807077. [13] SageMath, the Sage Mathematics Software System (Version 8.2), The Sage Developers, 2018, http://www.sagemath.org [14] The GAP Group, GAP – Groups: Algorithms, and Programming, Version 4.8.4; 2016., http://www.gap-system.org [15] A. Švob, Transitive distance-regular graphs from linear groups $L(3, q)$, q=2, 3, 4, 5, Trans. Comb., 9 (2020), 49-60.

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##### References:
 [1] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I, The user language, J. Symb. Comput., 24 (1997), 235-265.  doi: 10.1006/jsco.1996.0125. [2] A. E. Brouwer, A. M. Cohen and A. Neumaier, Distance-Regular Graphs, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-642-74341-2. [3] D. Crnković, M. Maksimović, B. G. Rodrigues and S. Rukavina, Self-orthogonal codes from the strongly regular graphs on up to 40 vertices, Adv. Math. Commun., 10 (2016), 555-582.  doi: 10.3934/amc.2016026. [4] D. Crnković, F. Pavese and A. Švob, On the PSU(4, 2)-invariant vertex-transitive strongly regular (216, 40, 4, 8) graph, Graphs Combin., 36 (2020), 503-513.  doi: 10.1007/s00373-020-02132-5. [5] D. Crnković, B. G. Rodrigues, S. Rukavina and L. Simčić, Self-orthogonal codes from orbit matrices of 2-designs, Adv. Math. Commun., 7 (2013), 161-174.  doi: 10.3934/amc.2013.7.161. [6] D. Crnković, S. Rukavina and A. Švob, Self-orthogonal codes from equitable partitions of association schemes, J. Algebraic Combin., 55 (2022), 157-171.  doi: 10.1007/s10801-021-01104-z. [7] D. Crnković, S. Rukavina and A. Švob, New strongly regular graphs from orthogonal groups $O^+(6, 2)$ and $O^-(6, 2)$, Discrete Math., 341 (2018), 2723-2728.  doi: 10.1016/j.disc.2018.06.029. [8] C. D. Godsil and W. J. Martin, Quotients of association schemes, J. Combin. Theory Ser. A, 69 (1995), 185-199.  doi: 10.1016/0097-3165(95)90050-0. [9] M. Grassl, Bounds on the minimum distance of linear codes and quantum codes, Accessed 24 October 2021, http://www.codetables.de. [10] A. Hanaki, Elementary functions for association schemes on GAP, 2013, http://math.shinshu-u.ac.jp/＄\sim＄hanaki/as/gap/association_scheme.pdf [11] M. Harada and V. Tonchev, Self-orthogonal codes from symmetric designs with fixed-point-free automorphisms, Discrete Math., 264 (2003), 81-90.  doi: 10.1016/S0012-365X(02)00553-8. [12] W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511807077. [13] SageMath, the Sage Mathematics Software System (Version 8.2), The Sage Developers, 2018, http://www.sagemath.org [14] The GAP Group, GAP – Groups: Algorithms, and Programming, Version 4.8.4; 2016., http://www.gap-system.org [15] A. Švob, Transitive distance-regular graphs from linear groups $L(3, q)$, q=2, 3, 4, 5, Trans. Comb., 9 (2020), 49-60.
The coefficients $p_{i,i}^k$ of adjacency matrices for the graph $\Gamma_{280}$
 $A_0$ $A_1$ $A_2$ $A_3$ $A_4$ $A_0 \cdot A_0$ 1 0 0 0 0 $A_1 \cdot A_1$ 9 0 1 0 0 $A_2 \cdot A_2$ 72 16 21 15 24 $A_3 \cdot A_3$ 144 48 84 68 80 $A_4 \cdot A_4$ 54 6 12 9 12
 $A_0$ $A_1$ $A_2$ $A_3$ $A_4$ $A_0 \cdot A_0$ 1 0 0 0 0 $A_1 \cdot A_1$ 9 0 1 0 0 $A_2 \cdot A_2$ 72 16 21 15 24 $A_3 \cdot A_3$ 144 48 84 68 80 $A_4 \cdot A_4$ 54 6 12 9 12
Self-orthogonal codes from the matrices $M_i^H$ for $\Gamma_{280}$
 i $H\leq P \Gamma L(3,4).Z_2$ orbit lengths code construction $3$ $I$ $280\times 1$ $[280,18,112]_2$ Cor. 1 $3$ $Z_3$ $10\times 1,90\times 3$ $[100,10,40]_2$ Cor. 1 $3$ $Z_3$ $7\times 1,91\times 3$ $[98,6,48]_2*$ Cor. 1 $3$ $Z_3$ $1\times 1,93\times 3$ $[94,6,38]_2$ Cor. 1 $3$ $Z_5$ $56\times 5$ $[56,2,24]_2$ Cor. 1 $3$ $E_9$ $1\times 1,9\times 3,28\times 9$ $[38,6,16]_2$ Cor. 1 $3$ $E_9$ $1\times 1,6\times 3,29\times 9$ $[36,2,18]_2$ Cor. 1 $3$ $E_9$ $1\times 1,31\times9$ $[32,2,16]_2$ Cor. 1 $3$ $Z_{15}$ $2\times 5,18\times 15$ $[20,2,8]_2$ Cor. 1 $3$ $E_9:Z_3$ $1\times 1,7\times 9,8\times 27$ $[16,2,8]_2$ Cor. 1 $3$ $Z_2$ $28\times 1,126\times 2$ $[28,6,12]_2*$, $[126,6,60]_2$ Thm. 2.5 $3$ $Z_2$ $10\times 1,135\times 2$ $[135,9,56]_2$ Thm. 2.5 $3$ $Z_2$ $10\times 1,135\times 2$ $[10,6,2]_2$, $[135,12,5]_2$ Thm. 2.5 $3$ $Z_2$ $8\times 1,136\times 2$ $[136,8,56]_2$ Thm. 2.5 $3$ $Z_4$ $4\times 1,2\times 2,68\times 4$, $4\times 2,68\times 4$ $[68,4,28]_2$ Thm. 2.5 $3$ $Z_4$ $6\times 1,1\times 2,68\times 4$ $[68,4,32]_2$ Thm. 2.5 $3$ $E_4$ $4\times 1,18\times 2,60\times 4$ $[18,3,6]_2$ $[60,3,30]_2$ Thm. 2.5 $3$ $E_4$ $4\times 1,26\times 2,56\times 4$ $[26,4,8]_2$, $[56,2,32]_2$ Thm. 2.5 $3$ $E_4$ $12\times 2,64\times 4$ $[64,2,28]_2$ Thm. 2.5 $3$ $E_4$ $12\times 2,64\times 4$ $[64,2,40]_2$ Thm. 2.5 $3$ $E_4$ $2\times 1,11\times 2,64\times 4$ $[64,4,30]_2$ Thm. 2.5 $4$ $Z_7$ $40\times 7$ $[40,12,12]_3$ Thm. 2.2 $4$ $Z_3$ $1\times 1,93\times 3$ $[93,27,15]_3$ Thm. 2.5 $4$ $Z_3$ $7\times 1,91\times 3$ $[91,28,18]_3$ Thm. 2.5 $4$ $Z_3$ $10\times 1,90\times 3$ $[90,25,18]_3$ Thm. 2.5 $4$ $E_9$ $1\times 1,31\times 9$ $[31,8,6]_3$ Thm. 2.5 $4$ $E_9$ $1\times 1,6\times 3,29\times 9$ $[29,9,6]_3$ Thm. 2.5 $4$ $E_9$ $1\times 1,9\times 3,28\times 9$ $[28,7,6]_3$ Thm. 2.5
 i $H\leq P \Gamma L(3,4).Z_2$ orbit lengths code construction $3$ $I$ $280\times 1$ $[280,18,112]_2$ Cor. 1 $3$ $Z_3$ $10\times 1,90\times 3$ $[100,10,40]_2$ Cor. 1 $3$ $Z_3$ $7\times 1,91\times 3$ $[98,6,48]_2*$ Cor. 1 $3$ $Z_3$ $1\times 1,93\times 3$ $[94,6,38]_2$ Cor. 1 $3$ $Z_5$ $56\times 5$ $[56,2,24]_2$ Cor. 1 $3$ $E_9$ $1\times 1,9\times 3,28\times 9$ $[38,6,16]_2$ Cor. 1 $3$ $E_9$ $1\times 1,6\times 3,29\times 9$ $[36,2,18]_2$ Cor. 1 $3$ $E_9$ $1\times 1,31\times9$ $[32,2,16]_2$ Cor. 1 $3$ $Z_{15}$ $2\times 5,18\times 15$ $[20,2,8]_2$ Cor. 1 $3$ $E_9:Z_3$ $1\times 1,7\times 9,8\times 27$ $[16,2,8]_2$ Cor. 1 $3$ $Z_2$ $28\times 1,126\times 2$ $[28,6,12]_2*$, $[126,6,60]_2$ Thm. 2.5 $3$ $Z_2$ $10\times 1,135\times 2$ $[135,9,56]_2$ Thm. 2.5 $3$ $Z_2$ $10\times 1,135\times 2$ $[10,6,2]_2$, $[135,12,5]_2$ Thm. 2.5 $3$ $Z_2$ $8\times 1,136\times 2$ $[136,8,56]_2$ Thm. 2.5 $3$ $Z_4$ $4\times 1,2\times 2,68\times 4$, $4\times 2,68\times 4$ $[68,4,28]_2$ Thm. 2.5 $3$ $Z_4$ $6\times 1,1\times 2,68\times 4$ $[68,4,32]_2$ Thm. 2.5 $3$ $E_4$ $4\times 1,18\times 2,60\times 4$ $[18,3,6]_2$ $[60,3,30]_2$ Thm. 2.5 $3$ $E_4$ $4\times 1,26\times 2,56\times 4$ $[26,4,8]_2$, $[56,2,32]_2$ Thm. 2.5 $3$ $E_4$ $12\times 2,64\times 4$ $[64,2,28]_2$ Thm. 2.5 $3$ $E_4$ $12\times 2,64\times 4$ $[64,2,40]_2$ Thm. 2.5 $3$ $E_4$ $2\times 1,11\times 2,64\times 4$ $[64,4,30]_2$ Thm. 2.5 $4$ $Z_7$ $40\times 7$ $[40,12,12]_3$ Thm. 2.2 $4$ $Z_3$ $1\times 1,93\times 3$ $[93,27,15]_3$ Thm. 2.5 $4$ $Z_3$ $7\times 1,91\times 3$ $[91,28,18]_3$ Thm. 2.5 $4$ $Z_3$ $10\times 1,90\times 3$ $[90,25,18]_3$ Thm. 2.5 $4$ $E_9$ $1\times 1,31\times 9$ $[31,8,6]_3$ Thm. 2.5 $4$ $E_9$ $1\times 1,6\times 3,29\times 9$ $[29,9,6]_3$ Thm. 2.5 $4$ $E_9$ $1\times 1,9\times 3,28\times 9$ $[28,7,6]_3$ Thm. 2.5
The coefficients $p_{i,i}^k$ of adjacency matrices for the graph $\Gamma_{216}$
 $A_0$ $A_1$ $A_2$ $A_0 \cdot A_0$ 1 0 0 $A_1 \cdot A_1$ 40 4 8 $A_2 \cdot A_2$ 175 140 142
 $A_0$ $A_1$ $A_2$ $A_0 \cdot A_0$ 1 0 0 $A_1 \cdot A_1$ 40 4 8 $A_2 \cdot A_2$ 175 140 142
Binary self-orthogonal codes from the matrices $M_1^H$ for the graph $\Gamma_{216}$, Cor. 1
 $H\leq Aut(\Gamma_{216})$ orbit lengths code $I$ $216\times 1$ $[ 216,60,32 ]$ $Z_3$ $72\times 3$ $[ 72, 18, 16 ]$ $Z_3$ $72\times 3$ $[ 72, 16, 16 ]$ $Z_3$ $9\times 1, 69\times 3$ $[ 78, 20, 16 ]$ $Z_5$ $1\times 1, 43\times 5$ $[ 44, 12, 8 ]$ $E_9$ $6\times 3, 22\times 9$ $[ 28, 4, 8 ]$ $E_9$ $9\times 3, 21\times 9$ $[ 30, 6, 8 ]$ $E_9$ $3\times 3, 23\times 9$ $[ 26, 4, 8 ]$ $Z_9$ $24\times 9$ $[ 24, 6, 8 ]$
 $H\leq Aut(\Gamma_{216})$ orbit lengths code $I$ $216\times 1$ $[ 216,60,32 ]$ $Z_3$ $72\times 3$ $[ 72, 18, 16 ]$ $Z_3$ $72\times 3$ $[ 72, 16, 16 ]$ $Z_3$ $9\times 1, 69\times 3$ $[ 78, 20, 16 ]$ $Z_5$ $1\times 1, 43\times 5$ $[ 44, 12, 8 ]$ $E_9$ $6\times 3, 22\times 9$ $[ 28, 4, 8 ]$ $E_9$ $9\times 3, 21\times 9$ $[ 30, 6, 8 ]$ $E_9$ $3\times 3, 23\times 9$ $[ 26, 4, 8 ]$ $Z_9$ $24\times 9$ $[ 24, 6, 8 ]$
Binary self-orthogonal codes from the matrices $M_1^H$ for the graph $\Gamma_{216}$, Thm. 2.5
 $H\leq Aut(\Gamma_{216})$ orbit lengths code $Z_2$ $6\times 1,105\times 2$ $[ 105, 26, 20 ]$ $Z_2$ $48 \times 1, 84\times 2$ $[ 48, 12, 12 ], [ 84, 24, 16 ]$ $Z_2$ $12\times 1,102\times 2$ $[ 102, 28, 16 ]$ $Z_2$ $10\times 1,103\times 2$ $[ 103, 30, 16 ]$ $Z_4$ $24\times 2, 42\times 4$ $[ 24, 4, 8 ], [ 42, 12, 8 ]$ $Z_4$ $6\times 2, 51\times 4$ $[ 51, 14, 8 ]$ $E_4$ $18\times 2, 45\times 4$ $[ 45, 10, 8 ]$ $E_4$ $12\times 1, 36\times 2, 33\times 4$ $[ 36, 8, 8 ], [ 33, 10, 8 ]$ $E_4$ $12\times 2, 48\times 4$ $[ 48, 12, 12 ]$ $E_4$ $36\times 2, 36\times 4$ $[ 36, 10, 8 ], [ 36, 4, 8 ]$ $E_4$ $4\times 1, 26\times 2, 40 \times 4$ $[ 26, 6, 6 ], [ 40, 10, 12 ]$ $E_4$ $2\times 1, 15\times 2, 46\times 4$ $[ 46, 12, 8 ]$ $Z_4$ $4\times 1, 4\times 2, 51\times 4$ $[ 51, 14, 8 ]$ $E_4$ $4\times 1, 10\times 2, 48\times 4$ $[ 48, 12, 12 ]$ $Z_4$ $2\times 1, 5\times 2, 51\times 4$ $[ 51, 14, 8 ]$ $E_4$ $2\times 1, 11\times 2, 48\times 4$ $[ 48, 12, 12 ]$ $E_4$ $2\times 1, 13\times 2, 47\times 4$ $[ 47, 14, 8 ]$ $E_4$ $4\times 1, 28\times 2, 39\times 4$ $[ 39, 12, 8 ],[ 28, 6, 6 ]$ $S_3$ $6\times 3, 33\times 6$ $[ 33, 4, 12 ]$ $Z_6$ $16\times 3, 28\times 6$ $[ 28, 8, 8 ]$ $Z_6$ $2\times 3, 35\times 6$ $[ 35, 6, 8 ]$ $S_3$ $10\times 3, 31\times 6$ $[ 31, 8, 8 ]$ $S_3$ $12\times 3, 30\times 6$ $[ 30, 6, 8 ]$ $Z_6$ $16\times 3, 28\times 6$ $[ 28, 6, 8 ], [ 16, 4, 8 ] *$ $S_3$ $10\times 3, 31\times 6$ $[ 31, 8, 8 ]$ $Z_6$ $4\times3, 34\times 6$ $[ 34, 8, 8 ]$ $S_3$ $10\times 3, 31\times 6$ $[ 31, 9, 8 ]$ $Q_8$ $12\times 4, 21\times 8$ $[ 21, 4, 8 ]$ $E_8$ $4\times 2, 24\times 4, 14\times 8$ $[ 14, 4, 4 ]$ $D_8$ $6\times 2, 21\times 4, 15\times 8$ $[ 15, 4, 8 ]*, [ 21, 4, 8 ]$ $E_8$ $18\times 2, 21\times 4, 12\times 8$ $[ 12, 4, 4 ]$ $Z_4\times Z_2$ $18\times 4, 18\times 8$ $[ 18, 4, 4 ]$ $E_8$ $4\times 2, 10\times 4, 21\times 8$ $[ 21, 6, 8 ]*$ $E_8$ $6\times 2, 9\times 4, 21\times 8$ $[ 21, 4, 8 ]$ $Z_4\times Z_2$ $2\times 2, 17\times 4, 18\times 8$ $[ 18, 4, 8 ] *$ $D_8$ $2\times 2, 11\times 4, 21\times 8$ $[ 21, 4, 8 ]$ $E_8$ $8\times 2, 14\times 4, 18\times 8$ $[ 18, 4, 8 ] *$ $Z_4\times Z_2$ $4\times 2, 16\times 4, 18\times 8$ $[ 18, 4, 8 ]*$ $D_8$ $4\times 1, 4\times 2, 21\times 4, 15\times 8$ $[ 21, 4, 8 ]$ $D_8$ $24\times 4, 15\times 8$ $[ 15, 4, 4 ]$ $E_8$ $10\times 2, 13\times 4, 18\times 8$ $[ 18, 4, 8 ]*$ $Z_4\times Z_2$ $2\times 2, 17\times 4, 18\times 8$ $[ 18, 4, 4 ]$ $D_8$ $2\times 2, 9\times 4, 22\times 8$ $[ 22, 6, 8 ]$ $D_8$ $4\times 2, 10\times 4, 21\times 8$ $[ 21, 4, 8 ]$ $D_8$ $2\times 2, 11\times 4, 21\times 8$ $[ 21, 6, 8 ]*$ $Z_8$ $12\times 4, 21\times 8$ $[ 21, 6, 4 ]$ $D_8$ $2\times 1, 5\times 2, 23\times 4, 14\times 8$ $[ 23, 4, 8 ]$ $E_8$ $2\times 1, 9\times 2, 19\times 4, 15\times 8$ $[ 19, 4, 4 ]$ $Z_4\times Z_2$ $2\times 2, 5\times 4, 24\times 8$ $[ 24, 6, 8 ]$ $D_8$ $2\times 2, 13\times 4, 20\times 8$ $[ 20, 4, 4 ]$ $D_8$ $4\times 2, 16\times 4, 18\times 8$ $[ 18, 6, 4 ]$ $Z_6\times Z_2$ $4\times 3, 12\times 6, 11\times 12$ $[ 12, 4, 4 ]$ $Z_{12}$ $8\times 6, 14\times 12$ $[ 14, 4, 4 ]$ $D_{12}$ $2\times 3, 9\times 6, 13\times 12$ $[ 13, 4, 4 ]$ $Z_{12}$ $2\times 6, 17\times 12$ $[ 17, 4, 4 ]$ $Z_3:Z_4$ $2\times 3, 1\times 6, 17\times 12$ $[ 17, 4, 4 ]$ $D_{12}$ $4\times 3, 12\times 6, 11\times 12$ $[ 11, 4, 4 ]$ $D_{12}$ $4\times 3, 12\times 6, 11\times 12$ $[ 11, 3, 4 ]$
 $H\leq Aut(\Gamma_{216})$ orbit lengths code $Z_2$ $6\times 1,105\times 2$ $[ 105, 26, 20 ]$ $Z_2$ $48 \times 1, 84\times 2$ $[ 48, 12, 12 ], [ 84, 24, 16 ]$ $Z_2$ $12\times 1,102\times 2$ $[ 102, 28, 16 ]$ $Z_2$ $10\times 1,103\times 2$ $[ 103, 30, 16 ]$ $Z_4$ $24\times 2, 42\times 4$ $[ 24, 4, 8 ], [ 42, 12, 8 ]$ $Z_4$ $6\times 2, 51\times 4$ $[ 51, 14, 8 ]$ $E_4$ $18\times 2, 45\times 4$ $[ 45, 10, 8 ]$ $E_4$ $12\times 1, 36\times 2, 33\times 4$ $[ 36, 8, 8 ], [ 33, 10, 8 ]$ $E_4$ $12\times 2, 48\times 4$ $[ 48, 12, 12 ]$ $E_4$ $36\times 2, 36\times 4$ $[ 36, 10, 8 ], [ 36, 4, 8 ]$ $E_4$ $4\times 1, 26\times 2, 40 \times 4$ $[ 26, 6, 6 ], [ 40, 10, 12 ]$ $E_4$ $2\times 1, 15\times 2, 46\times 4$ $[ 46, 12, 8 ]$ $Z_4$ $4\times 1, 4\times 2, 51\times 4$ $[ 51, 14, 8 ]$ $E_4$ $4\times 1, 10\times 2, 48\times 4$ $[ 48, 12, 12 ]$ $Z_4$ $2\times 1, 5\times 2, 51\times 4$ $[ 51, 14, 8 ]$ $E_4$ $2\times 1, 11\times 2, 48\times 4$ $[ 48, 12, 12 ]$ $E_4$ $2\times 1, 13\times 2, 47\times 4$ $[ 47, 14, 8 ]$ $E_4$ $4\times 1, 28\times 2, 39\times 4$ $[ 39, 12, 8 ],[ 28, 6, 6 ]$ $S_3$ $6\times 3, 33\times 6$ $[ 33, 4, 12 ]$ $Z_6$ $16\times 3, 28\times 6$ $[ 28, 8, 8 ]$ $Z_6$ $2\times 3, 35\times 6$ $[ 35, 6, 8 ]$ $S_3$ $10\times 3, 31\times 6$ $[ 31, 8, 8 ]$ $S_3$ $12\times 3, 30\times 6$ $[ 30, 6, 8 ]$ $Z_6$ $16\times 3, 28\times 6$ $[ 28, 6, 8 ], [ 16, 4, 8 ] *$ $S_3$ $10\times 3, 31\times 6$ $[ 31, 8, 8 ]$ $Z_6$ $4\times3, 34\times 6$ $[ 34, 8, 8 ]$ $S_3$ $10\times 3, 31\times 6$ $[ 31, 9, 8 ]$ $Q_8$ $12\times 4, 21\times 8$ $[ 21, 4, 8 ]$ $E_8$ $4\times 2, 24\times 4, 14\times 8$ $[ 14, 4, 4 ]$ $D_8$ $6\times 2, 21\times 4, 15\times 8$ $[ 15, 4, 8 ]*, [ 21, 4, 8 ]$ $E_8$ $18\times 2, 21\times 4, 12\times 8$ $[ 12, 4, 4 ]$ $Z_4\times Z_2$ $18\times 4, 18\times 8$ $[ 18, 4, 4 ]$ $E_8$ $4\times 2, 10\times 4, 21\times 8$ $[ 21, 6, 8 ]*$ $E_8$ $6\times 2, 9\times 4, 21\times 8$ $[ 21, 4, 8 ]$ $Z_4\times Z_2$ $2\times 2, 17\times 4, 18\times 8$ $[ 18, 4, 8 ] *$ $D_8$ $2\times 2, 11\times 4, 21\times 8$ $[ 21, 4, 8 ]$ $E_8$ $8\times 2, 14\times 4, 18\times 8$ $[ 18, 4, 8 ] *$ $Z_4\times Z_2$ $4\times 2, 16\times 4, 18\times 8$ $[ 18, 4, 8 ]*$ $D_8$ $4\times 1, 4\times 2, 21\times 4, 15\times 8$ $[ 21, 4, 8 ]$ $D_8$ $24\times 4, 15\times 8$ $[ 15, 4, 4 ]$ $E_8$ $10\times 2, 13\times 4, 18\times 8$ $[ 18, 4, 8 ]*$ $Z_4\times Z_2$ $2\times 2, 17\times 4, 18\times 8$ $[ 18, 4, 4 ]$ $D_8$ $2\times 2, 9\times 4, 22\times 8$ $[ 22, 6, 8 ]$ $D_8$ $4\times 2, 10\times 4, 21\times 8$ $[ 21, 4, 8 ]$ $D_8$ $2\times 2, 11\times 4, 21\times 8$ $[ 21, 6, 8 ]*$ $Z_8$ $12\times 4, 21\times 8$ $[ 21, 6, 4 ]$ $D_8$ $2\times 1, 5\times 2, 23\times 4, 14\times 8$ $[ 23, 4, 8 ]$ $E_8$ $2\times 1, 9\times 2, 19\times 4, 15\times 8$ $[ 19, 4, 4 ]$ $Z_4\times Z_2$ $2\times 2, 5\times 4, 24\times 8$ $[ 24, 6, 8 ]$ $D_8$ $2\times 2, 13\times 4, 20\times 8$ $[ 20, 4, 4 ]$ $D_8$ $4\times 2, 16\times 4, 18\times 8$ $[ 18, 6, 4 ]$ $Z_6\times Z_2$ $4\times 3, 12\times 6, 11\times 12$ $[ 12, 4, 4 ]$ $Z_{12}$ $8\times 6, 14\times 12$ $[ 14, 4, 4 ]$ $D_{12}$ $2\times 3, 9\times 6, 13\times 12$ $[ 13, 4, 4 ]$ $Z_{12}$ $2\times 6, 17\times 12$ $[ 17, 4, 4 ]$ $Z_3:Z_4$ $2\times 3, 1\times 6, 17\times 12$ $[ 17, 4, 4 ]$ $D_{12}$ $4\times 3, 12\times 6, 11\times 12$ $[ 11, 4, 4 ]$ $D_{12}$ $4\times 3, 12\times 6, 11\times 12$ $[ 11, 3, 4 ]$
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