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November  2020, 14(4): 591-602. doi: 10.3934/amc.2020032

Self-orthogonal codes from orbit matrices of Seidel and Laplacian matrices of strongly regular graphs

Dean Crnković 1, , Ronan Egan 2, and  Andrea Švob 1,,

1. 

Department of Mathematics, University of Rijeka, Croatia
2. 

De Brún Centre for Mathematics, School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway

* Corresponding author: A. Švob

Received  April 2018 Revised  April 2019 Published  November 2020 Early access  November 2019

Fund Project: D. Crnković and A. Švob were supported by Croatian Science Foundation under the project 6732. R. Egan was supported by the Irish Research Council (Government of Ireland Postdoctoral Fellowship, GOIPD/2018/304)

In this paper we introduce the notion of orbit matrices of integer matrices such as Seidel and Laplacian matrices of some strongly regular graphs with respect to their permutation automorphism groups. We further show that under certain conditions these orbit matrices yield self-orthogonal codes over finite fields $ \mathbb{F}_q $, where $ q $ is a prime power and over finite rings $ \mathbb{Z}_m $. As a case study, we construct codes from orbit matrices of Seidel, Laplacian and signless Laplacian matrices of strongly regular graphs. In particular, we construct self-orthogonal codes from orbit matrices of Seidel and Laplacian matrices of the Higman-Sims and McLaughlin graphs.

Keywords: Strongly regular graph, Seidel matrix, Laplacian matrix, orbit matrix, self-orthogonal code.
Mathematics Subject Classification: Primary: 05E30, 94B05; Secondary: 05E18.
Citation: Dean Crnković, Ronan Egan, Andrea Švob. Self-orthogonal codes from orbit matrices of Seidel and Laplacian matrices of strongly regular graphs. Advances in Mathematics of Communications, 2020, 14 (4) : 591-602. doi: 10.3934/amc.2020032
References:
[1]

M. Behbahani and C. Lam, Strongly regular graphs with non-trivial automorphisms, Discrete Math., 311 (2011), 132-144.  doi: 10.1016/j.disc.2010.10.005.  Google Scholar

[2]

W. Bosma, J. Cannon and C. Playoust, The Magma algebra system Ⅰ: The user language, J. Symbolic Comput., 24 (1997), 235-265.  doi: 10.1006/jsco.1996.0125.  Google Scholar

[3]

A. E. Brouwer, A. M. Cohen and A. Neumaier, Distance-Regular Graphs, Results in Mathematics and Related Areas, 18, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-642-74341-2.  Google Scholar

[4]

A. E. Brouwer, Strongly regular graphs, in Handbook of Combinatorial Designs, Chapman & Hall/CRC, Boca Raton, FL, 2007,852-868.  Google Scholar

[5]

A. E. Brouwer, Parameters of strongly regular graphs, Available from: http://www.win.tue.nl/~aeb/graphs/srg/srgtab.html. Google Scholar

[6]

A. E. Brouwer and W. H. Haemers, Spectra of Graphs, Universitext, Springer, New York, 2012. doi: 10.1007/978-1-4614-1939-6.  Google Scholar

[7]

D. Crnković, R. Egan and A. Švob, Orbit matrices of Hadamard matrices and related codes, Discrete Math., 341 (2018), 1199-1209.  doi: 10.1016/j.disc.2018.01.018.  Google Scholar

[8]

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.  Google Scholar

[9]

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.  Google Scholar

[10]

S. T. Dougherty, Algebraic Coding Theory Over Finite Commutative Rings, SpringerBriefs in Mathematics, Springer, Cham, 2017. doi: 10.1007/978-3-319-59806-2.  Google Scholar

[11]

M. Grassl, Bounds on the minimum distance of linear codes and quantum codes, Available from: http://www.codetables.de. Google Scholar

[12]

W. H. Haemers, R. Peeters and J. M. van Rijckevorsel, Binary codes of strongly regular graphs, Des. Codes Cryptogr., 17 (1999), 187-209.  doi: 10.1023/A:1026479210284.  Google Scholar

[13]

W. H. Haemers and E. Spence, Enumeration of cospectral graphs, European J. Combin., 25 (2004), 199-211.  doi: 10.1016/S0195-6698(03)00100-8.  Google Scholar

[14]

M. Harada, Note on the residue codes of self-dual $\mathbb{Z}_4$-codes having large minimum Lee weights, Adv. Math. Commun., 10 (2016), 695-706.  doi: 10.3934/amc.2016035.  Google Scholar

[15]

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.  Google Scholar

[16] W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511807077.  Google Scholar
[17]

J. D. Key, T. P. McDonough and V. C. Mavron, Improved partial permutation decoding for Reed-Muller codes, Discrete Math., 340 (2017), 722-728.  doi: 10.1016/j.disc.2016.11.031.  Google Scholar

[18]

F. J. MacWilliams, Permutation decoding of systematic codes, Bell System Tech. J., 43 (1964), 485-505.  doi: 10.1002/j.1538-7305.1964.tb04075.x.  Google Scholar

[19]

E. Spence, Strongly regular graphs on at most 64 vertices, Available from: http://www.maths.gla.ac.uk/~es/srgraphs.php. Google Scholar

[20]

F. Szöllősi and P. R. J. Östergård, Enumeration of Seidel matrices, European J. Combin., 69 (2018), 169-184.  doi: 10.1016/j.ejc.2017.10.009.  Google Scholar

[21]

V. D. Tonchev, Combinatorial Configurations: Designs, Codes, Graphs, Pitman Monographs and Surveys in Pure and Applied Mathematics, 40, John Wiley & Sons, Inc., New York, 1988.  Google Scholar

show all references

References:
[1]

M. Behbahani and C. Lam, Strongly regular graphs with non-trivial automorphisms, Discrete Math., 311 (2011), 132-144.  doi: 10.1016/j.disc.2010.10.005.  Google Scholar

[2]

W. Bosma, J. Cannon and C. Playoust, The Magma algebra system Ⅰ: The user language, J. Symbolic Comput., 24 (1997), 235-265.  doi: 10.1006/jsco.1996.0125.  Google Scholar

[3]

A. E. Brouwer, A. M. Cohen and A. Neumaier, Distance-Regular Graphs, Results in Mathematics and Related Areas, 18, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-642-74341-2.  Google Scholar

[4]

A. E. Brouwer, Strongly regular graphs, in Handbook of Combinatorial Designs, Chapman & Hall/CRC, Boca Raton, FL, 2007,852-868.  Google Scholar

[5]

A. E. Brouwer, Parameters of strongly regular graphs, Available from: http://www.win.tue.nl/~aeb/graphs/srg/srgtab.html. Google Scholar

[6]

A. E. Brouwer and W. H. Haemers, Spectra of Graphs, Universitext, Springer, New York, 2012. doi: 10.1007/978-1-4614-1939-6.  Google Scholar

[7]

D. Crnković, R. Egan and A. Švob, Orbit matrices of Hadamard matrices and related codes, Discrete Math., 341 (2018), 1199-1209.  doi: 10.1016/j.disc.2018.01.018.  Google Scholar

[8]

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.  Google Scholar

[9]

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.  Google Scholar

[10]

S. T. Dougherty, Algebraic Coding Theory Over Finite Commutative Rings, SpringerBriefs in Mathematics, Springer, Cham, 2017. doi: 10.1007/978-3-319-59806-2.  Google Scholar

[11]

M. Grassl, Bounds on the minimum distance of linear codes and quantum codes, Available from: http://www.codetables.de. Google Scholar

[12]

W. H. Haemers, R. Peeters and J. M. van Rijckevorsel, Binary codes of strongly regular graphs, Des. Codes Cryptogr., 17 (1999), 187-209.  doi: 10.1023/A:1026479210284.  Google Scholar

[13]

W. H. Haemers and E. Spence, Enumeration of cospectral graphs, European J. Combin., 25 (2004), 199-211.  doi: 10.1016/S0195-6698(03)00100-8.  Google Scholar

[14]

M. Harada, Note on the residue codes of self-dual $\mathbb{Z}_4$-codes having large minimum Lee weights, Adv. Math. Commun., 10 (2016), 695-706.  doi: 10.3934/amc.2016035.  Google Scholar

[15]

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.  Google Scholar

[16] W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511807077.  Google Scholar
[17]

J. D. Key, T. P. McDonough and V. C. Mavron, Improved partial permutation decoding for Reed-Muller codes, Discrete Math., 340 (2017), 722-728.  doi: 10.1016/j.disc.2016.11.031.  Google Scholar

[18]

F. J. MacWilliams, Permutation decoding of systematic codes, Bell System Tech. J., 43 (1964), 485-505.  doi: 10.1002/j.1538-7305.1964.tb04075.x.  Google Scholar

[19]

E. Spence, Strongly regular graphs on at most 64 vertices, Available from: http://www.maths.gla.ac.uk/~es/srgraphs.php. Google Scholar

[20]

F. Szöllősi and P. R. J. Östergård, Enumeration of Seidel matrices, European J. Combin., 69 (2018), 169-184.  doi: 10.1016/j.ejc.2017.10.009.  Google Scholar

[21]

V. D. Tonchev, Combinatorial Configurations: Designs, Codes, Graphs, Pitman Monographs and Surveys in Pure and Applied Mathematics, 40, John Wiley & Sons, Inc., New York, 1988.  Google Scholar

Table 1.  Self-orthogonal codes constructed from Seidel matrices of SRGs
Graph $ C $ $ \mathrm{Dual}(C) $ $ |\mathrm{Aut}(C)| $
$ \mathcal{G}_1^1 $ $ [10,4,6]_3* $ $ [10,6,4]_3* $ 2880
$ \mathcal{G}_2^1 $ $ [26,11,8]_5 $ $ [26,15,6]_5 $ 576
$ \mathcal{G}_2^3 $ $ [26,12,8]_5 $ $ [26,14,6]_5 $ 48
$ \mathcal{G}_2^6 $ $ [26,12,8]_5 $ $ [26,14,6]_5 $ 312
$ \mathcal{G}_2^8 $ $ [26,9,14]_5* $ $ [28,17,6]_5 $ 124800
$ \mathcal{G}_3^1 $ $ [28,7,12]_3 $ $ [28,21,4]_3* $ $ 2^{10}\cdot 3^4\cdot 5^1 \cdot 7^1 $
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Graph $ C $ $ \mathrm{Dual}(C) $ $ |\mathrm{Aut}(C)| $
$ \mathcal{G}_1^1 $ $ [10,4,6]_3* $ $ [10,6,4]_3* $ 2880
$ \mathcal{G}_2^1 $ $ [26,11,8]_5 $ $ [26,15,6]_5 $ 576
$ \mathcal{G}_2^3 $ $ [26,12,8]_5 $ $ [26,14,6]_5 $ 48
$ \mathcal{G}_2^6 $ $ [26,12,8]_5 $ $ [26,14,6]_5 $ 312
$ \mathcal{G}_2^8 $ $ [26,9,14]_5* $ $ [28,17,6]_5 $ 124800
$ \mathcal{G}_3^1 $ $ [28,7,12]_3 $ $ [28,21,4]_3* $ $ 2^{10}\cdot 3^4\cdot 5^1 \cdot 7^1 $
Table 2.  Self-orthogonal codes constructed from orbit matrices of the Seidel matrix of $ \mathcal{G}_1 $
$ G \leq \mathrm{Aut}({\mathcal{G}_{1}}) $ $ C $ $ \mathrm{Dual}(C) $ $ |\mathrm{Aut}(C)| $
$ Z_3 $ $ [69,21,15]_3 $ $ [69,48,5]_3 $ 80
$ Z_{13} $ $ [16,4,9]_3* $ $ [16,12,1]_3 $ 2880
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$ G \leq \mathrm{Aut}({\mathcal{G}_{1}}) $ $ C $ $ \mathrm{Dual}(C) $ $ |\mathrm{Aut}(C)| $
$ Z_3 $ $ [69,21,15]_3 $ $ [69,48,5]_3 $ 80
$ Z_{13} $ $ [16,4,9]_3* $ $ [16,12,1]_3 $ 2880
Table 3.  Self-orthogonal codes constructed from orbit matrices of the Seidel matrix of $ \mathcal{G}_2 $
$ G \leq \mathrm{Aut}({\mathcal{G}_{2}}) $ $ C $ $ \mathrm{Dual}(C) $ $ |\mathrm{Aut}(C)| $
$ Z_3 $ $ [10,4,6]_3* $ $ [10,6,4]_3* $ 2880
$ Z_3 $ $ [36,14,12]_3 $ $ [36,22,6]_3 $ 2903040
$ Z_3 $ $ [28,7,12]_3 $ $ [28,21,4]_3* $ 2903040
$ Z_3 $ $ [42,15,12]_3 $ $ [42,27,4]_3 $ 17280
$ Z_3 $ $ [45,15,12]_3 $ $ [45,30,4]_3 $ 5184
$ Z_{17} $ $ [8,2,6]_3* $ $ [8,6,2]_3* $ 768
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$ G \leq \mathrm{Aut}({\mathcal{G}_{2}}) $ $ C $ $ \mathrm{Dual}(C) $ $ |\mathrm{Aut}(C)| $
$ Z_3 $ $ [10,4,6]_3* $ $ [10,6,4]_3* $ 2880
$ Z_3 $ $ [36,14,12]_3 $ $ [36,22,6]_3 $ 2903040
$ Z_3 $ $ [28,7,12]_3 $ $ [28,21,4]_3* $ 2903040
$ Z_3 $ $ [42,15,12]_3 $ $ [42,27,4]_3 $ 17280
$ Z_3 $ $ [45,15,12]_3 $ $ [45,30,4]_3 $ 5184
$ Z_{17} $ $ [8,2,6]_3* $ $ [8,6,2]_3* $ 768
Table 4.  Self-orthogonal codes constructed from orbit matrices of the Laplacian matrix of $ \mathcal{G}_3 $
$ G \leq \mathrm{Aut}({\mathcal{G}_{3}}) $ $ C $ $ \mathrm{Dual}(C) $ $ |\mathrm{Aut}(C)| $
$ Z_3 $ $ [12,2,3]_3 $ $ [12,10,2]_3* $ $ 2^{10}\cdot 3^5\cdot 5\cdot 7 $
$ Z_3 $ $ [15,5,6]_3 $ $ [15,10,3]_3 $ 10
$ Z_3 $ $ [40,10,18]_3 $ $ [40,30,4]_3 $ $ 2^9\cdot 3^6\cdot 5\cdot 13 $
$ Z_3 $ $ [45,15,12]_3 $ $ [45,30,6]_3 $ 103680
$ Z_3 $ $ [51,13,12]_3 $ $ [51,38,4]_3 $ 10368
$ Z_3 $ $ [52,13,12]_3 $ $ [52,39,3]_3 $ 5184
$ Z_3 $ $ [53,14,12]_3 $ $ [53,39,4]_3 $ 864
$ Z_5 $ $ [33,9,12]_3 $ $ [33,24,2]_3 $ 96
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$ G \leq \mathrm{Aut}({\mathcal{G}_{3}}) $ $ C $ $ \mathrm{Dual}(C) $ $ |\mathrm{Aut}(C)| $
$ Z_3 $ $ [12,2,3]_3 $ $ [12,10,2]_3* $ $ 2^{10}\cdot 3^5\cdot 5\cdot 7 $
$ Z_3 $ $ [15,5,6]_3 $ $ [15,10,3]_3 $ 10
$ Z_3 $ $ [40,10,18]_3 $ $ [40,30,4]_3 $ $ 2^9\cdot 3^6\cdot 5\cdot 13 $
$ Z_3 $ $ [45,15,12]_3 $ $ [45,30,6]_3 $ 103680
$ Z_3 $ $ [51,13,12]_3 $ $ [51,38,4]_3 $ 10368
$ Z_3 $ $ [52,13,12]_3 $ $ [52,39,3]_3 $ 5184
$ Z_3 $ $ [53,14,12]_3 $ $ [53,39,4]_3 $ 864
$ Z_5 $ $ [33,9,12]_3 $ $ [33,24,2]_3 $ 96
Table 5.  Self-orthogonal codes constructed from orbit matrices of Laplace matrix of $ \mathcal{G}_4 $
$ G \leq \mathrm{Aut}({\mathcal{G}_{4}}) $ $ C $ $ \mathrm{Dual}(C) $ $ |\mathrm{Aut}(C)| $
$ Z_2 $ $ [12,2,6]_2 $ $ [12,10,2]_2* $ 1036800
$ Z_2 $ $ [14,7,4]_2* $ $ [14,7,4]_2* $ 56448
$ Z_2 $ $ [40,14,8]_2 $ $ [40,26,4]_2 $ 3932160
$ Z_2 $ $ [120,24,24]_2 $ $ [120,96,5]_2 $ 1920
$ Z_2 $ $ [133,27,24]_2 $ $ [133,106,6]_2 $ 336
$ Z_2 $ $ [134,30,24]_2 $ $ [134,104,5]_2 $ 240
$ Z_4 $ $ [16,6,6]_2* $ $ [16,10,4]_2* $ 11520
$ Z_4 $ $ [18,3,6]_2 $ $ [18,15,2]_2* $ $ 2^{13}\cdot 3^{7}\cdot 5^3 $
$ Z_4 $ $ [18,4,8]_2* $ $ [18,14,2]_2* $ 36864
$ Z_4 $ $ [36,6,8]_2 $ $ [36,30,2]_2 $ $ 2^{31}\cdot 3^{13} $
$ Z_4 $ $ [48,8,16_2 $ $ [48,40,4]_2* $ 69120
$ Z_4 $ $ [60,12,12]_2 $ $ [60,48,4]_2 $ 49152
$ Z_4 $ $ [61,13,16]_2 $ $ [61,48,4]_2 $ 17280
$ Z_5 $ $ [56,10,16]_2 $ $ [56,46,2]_2 $ 1440
$ Z_7 $ $ [40,8,8]_2 $ $ [40,32,2]_2 $ 393216
$ Z_7 $ $ [40,6,14]_5 $ $ [40,34,2]_5 $ 3072
$ Z_5 $ $ [56,8,20]_5 $ $ [56,48,2]_5 $ 115200
$ Z_5 $ $ [54,8,20]_5 $ $ [56,48,2]_5 $ 91729428480
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$ G \leq \mathrm{Aut}({\mathcal{G}_{4}}) $ $ C $ $ \mathrm{Dual}(C) $ $ |\mathrm{Aut}(C)| $
$ Z_2 $ $ [12,2,6]_2 $ $ [12,10,2]_2* $ 1036800
$ Z_2 $ $ [14,7,4]_2* $ $ [14,7,4]_2* $ 56448
$ Z_2 $ $ [40,14,8]_2 $ $ [40,26,4]_2 $ 3932160
$ Z_2 $ $ [120,24,24]_2 $ $ [120,96,5]_2 $ 1920
$ Z_2 $ $ [133,27,24]_2 $ $ [133,106,6]_2 $ 336
$ Z_2 $ $ [134,30,24]_2 $ $ [134,104,5]_2 $ 240
$ Z_4 $ $ [16,6,6]_2* $ $ [16,10,4]_2* $ 11520
$ Z_4 $ $ [18,3,6]_2 $ $ [18,15,2]_2* $ $ 2^{13}\cdot 3^{7}\cdot 5^3 $
$ Z_4 $ $ [18,4,8]_2* $ $ [18,14,2]_2* $ 36864
$ Z_4 $ $ [36,6,8]_2 $ $ [36,30,2]_2 $ $ 2^{31}\cdot 3^{13} $
$ Z_4 $ $ [48,8,16_2 $ $ [48,40,4]_2* $ 69120
$ Z_4 $ $ [60,12,12]_2 $ $ [60,48,4]_2 $ 49152
$ Z_4 $ $ [61,13,16]_2 $ $ [61,48,4]_2 $ 17280
$ Z_5 $ $ [56,10,16]_2 $ $ [56,46,2]_2 $ 1440
$ Z_7 $ $ [40,8,8]_2 $ $ [40,32,2]_2 $ 393216
$ Z_7 $ $ [40,6,14]_5 $ $ [40,34,2]_5 $ 3072
$ Z_5 $ $ [56,8,20]_5 $ $ [56,48,2]_5 $ 115200
$ Z_5 $ $ [54,8,20]_5 $ $ [56,48,2]_5 $ 91729428480
Table 6.  Self-orthogonal codes constructed from orbit matrices of the signless Laplacian matrix of $ \mathcal{G}_5 $
$ G \leq \mathrm{Aut}({\mathcal{G}_{5}}) $ $ C $ $ \mathrm{Dual}(C) $ $ |\mathrm{Aut}(C)| $
$ Z_2 $ $ [16,3,8]_2* $ $ [16,13,2]_2* $ $ 2^{15}\cdot 3^5 $
$ Z_2 $ $ [32,5,16]_2* $ $ [32,27,2]_2* $ $ 2^{26}\cdot 3^2\cdot 5^1\cdot 7^1 $
$ Z_2 $ $ [40,5,16]_2 $ $ [40,35,2]_2* $ $ 2^{34}\cdot 3^{12}\cdot 5^1 $
$ Z_2 $ $ [60,3,32]_2 $ $ [60,57,2]_2 $ $ 2^{55}\cdot 3^{18}\cdot 5^8\cdot 7^7\cdot 11^1 $
$ Z_2 $ $ [64,4,32]_2* $ $ [64,60,2]_2* $ $ 2^{55}\cdot 3^{17}\cdot 5^1\cdot 7^2 $
$ Z_2 $ $ [64,7,32]_2* $ $ [64,57,4]_2* $ $ 2^{21}\cdot 3^4\cdot 5^1\cdot 7^2\cdot 31^1 $
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$ G \leq \mathrm{Aut}({\mathcal{G}_{5}}) $ $ C $ $ \mathrm{Dual}(C) $ $ |\mathrm{Aut}(C)| $
$ Z_2 $ $ [16,3,8]_2* $ $ [16,13,2]_2* $ $ 2^{15}\cdot 3^5 $
$ Z_2 $ $ [32,5,16]_2* $ $ [32,27,2]_2* $ $ 2^{26}\cdot 3^2\cdot 5^1\cdot 7^1 $
$ Z_2 $ $ [40,5,16]_2 $ $ [40,35,2]_2* $ $ 2^{34}\cdot 3^{12}\cdot 5^1 $
$ Z_2 $ $ [60,3,32]_2 $ $ [60,57,2]_2 $ $ 2^{55}\cdot 3^{18}\cdot 5^8\cdot 7^7\cdot 11^1 $
$ Z_2 $ $ [64,4,32]_2* $ $ [64,60,2]_2* $ $ 2^{55}\cdot 3^{17}\cdot 5^1\cdot 7^2 $
$ Z_2 $ $ [64,7,32]_2* $ $ [64,57,4]_2* $ $ 2^{21}\cdot 3^4\cdot 5^1\cdot 7^2\cdot 31^1 $
Table 7.  Self-orthogonal codes constructed from orbit matrices of the Seidel matrix of $ \mathcal{G}_6 $
$ G \leq \mathrm{Aut}({\mathcal{G}_{6}}) $ $ C $ $ \mathrm{Dual}(C) $ $ |\mathrm{Aut}(C)| $
$ Z_5 $ $ [19,4,10]_5 $ $ [19,15,2]_5 $ 1920
$ Z_5 $ $ [20,3,10]_5 $ $ [20,17,2]_5 $ $ 2^{17}\cdot 3^5\cdot 5^4 $
$ Z_5 $ $ [20,4,14]_5* $ $ [20,16,4]_5* $ 960
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$ G \leq \mathrm{Aut}({\mathcal{G}_{6}}) $ $ C $ $ \mathrm{Dual}(C) $ $ |\mathrm{Aut}(C)| $
$ Z_5 $ $ [19,4,10]_5 $ $ [19,15,2]_5 $ 1920
$ Z_5 $ $ [20,3,10]_5 $ $ [20,17,2]_5 $ $ 2^{17}\cdot 3^5\cdot 5^4 $
$ Z_5 $ $ [20,4,14]_5* $ $ [20,16,4]_5* $ 960
Table 8.  Self-orthogonal codes constructed from orbit matrices of the Laplacian matrix of $ \mathcal{G}_7 $
$ G \leq \mathrm{Aut}({\mathcal{G}_{7}}) $ $ C $ $ \mathrm{Dual}(C) $ $ |\mathrm{Aut}(C)| $
$ Z_5 $ $ [54,4,20]_5 $ $ [54,50,2]_5 $ $ 2^{24}\cdot 3^{7}\cdot 5^{1} $
$ Z_5 $ $ [55,3,20]_5 $ $ [55,52,2]_5* $ $ 2^{44}\cdot 3^{17}\cdot 5^{12}\cdot 7^3\cdot 11^2 \cdot 13\cdot 17\cdot 19\cdot 23 $
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$ G \leq \mathrm{Aut}({\mathcal{G}_{7}}) $ $ C $ $ \mathrm{Dual}(C) $ $ |\mathrm{Aut}(C)| $
$ Z_5 $ $ [54,4,20]_5 $ $ [54,50,2]_5 $ $ 2^{24}\cdot 3^{7}\cdot 5^{1} $
$ Z_5 $ $ [55,3,20]_5 $ $ [55,52,2]_5* $ $ 2^{44}\cdot 3^{17}\cdot 5^{12}\cdot 7^3\cdot 11^2 \cdot 13\cdot 17\cdot 19\cdot 23 $
Table 9.  Self-dual codes over $ \mathbb{Z}_4 $ constructed from orbit matrices of signless Laplacian matrices of SRGs $ \mathcal{G}_5 $ and $ \mathcal{G}_8 $
Graph $ C $ $ d_H(C) $, $ d_E(C) $, $ d_L(C) $ Type
$ \mathcal{G}_5 $ $ ((8,4^1 2^{6})) $ 2, 8, 4 Ⅱ
$ \mathcal{G}_5 $ $ ((16,4^3 2^{10})) $ 2, 8, 4 Ⅱ
$ \mathcal{G}_5 $ $ ((32,4^5 2^{22})) $ 2, 8, 4 Ⅱ
$ \mathcal{G}_5 $ $ ((36,4^1 2^{34})) $ 2, 8, 4 Ⅰ
$ \mathcal{G}_5 $ $ ((40,4^5 2^{30})) $ 2, 8, 4 Ⅱ
$ \mathcal{G}_8 $ $ ((24,4^1 2^{22})) $ 2, 8, 4 Ⅱ
$ \mathcal{G}_8 $ $ ((24,4^5 2^{14})) $ 2, 8, 4 Ⅱ
$ \mathcal{G}_8 $ $ ((28,4^1 2^{26})) $ 2, 8, 4 Ⅰ
$ \mathcal{G}_8 $ $ ((30,4^0 2^{30})) $ 1, 4, 2 Ⅰ
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Graph $ C $ $ d_H(C) $, $ d_E(C) $, $ d_L(C) $ Type
$ \mathcal{G}_5 $ $ ((8,4^1 2^{6})) $ 2, 8, 4 Ⅱ
$ \mathcal{G}_5 $ $ ((16,4^3 2^{10})) $ 2, 8, 4 Ⅱ
$ \mathcal{G}_5 $ $ ((32,4^5 2^{22})) $ 2, 8, 4 Ⅱ
$ \mathcal{G}_5 $ $ ((36,4^1 2^{34})) $ 2, 8, 4 Ⅰ
$ \mathcal{G}_5 $ $ ((40,4^5 2^{30})) $ 2, 8, 4 Ⅱ
$ \mathcal{G}_8 $ $ ((24,4^1 2^{22})) $ 2, 8, 4 Ⅱ
$ \mathcal{G}_8 $ $ ((24,4^5 2^{14})) $ 2, 8, 4 Ⅱ
$ \mathcal{G}_8 $ $ ((28,4^1 2^{26})) $ 2, 8, 4 Ⅰ
$ \mathcal{G}_8 $ $ ((30,4^0 2^{30})) $ 1, 4, 2 Ⅰ
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