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

Dean Crnković , , Sanja Rukavina and  Andrea Švob

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.

Keywords: Distance-regular graph, adjacency matrix, equitable partition, self-orthogonal code.
Mathematics Subject Classification: Primary: 05E30, 94B05; Secondary: 05E18, 94B60.
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. 

show all references

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. 

Table 1.  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
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$ 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
Table 2.  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
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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
Table 3.  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
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$ 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
Table 4.  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 ] $
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$ 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 ] $
Table 5.  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 ] $
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$ 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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