# American Institute of Mathematical Sciences

August  2016, 10(3): 555-582. doi: 10.3934/amc.2016026

## Self-orthogonal codes from the strongly regular graphs on up to 40 vertices

 1 Department of Mathematics, University of Rijeka, Radmile Matejčić 2, 51000 Rijeka, Croatia 2 School of Mathematics, Statistics & Computer Science, University of KwaZulu-Natal, Durban 4000, South Africa

Received  May 2015 Revised  June 2016 Published  August 2016

This paper outlines a method for constructing self-orthogonal codes from orbit matrices of strongly regular graphs admitting an automorphism group $G$ which acts with orbits of length $w$, where $w$ divides $|G|$. We apply this method to construct self-orthogonal codes from orbit matrices of the strongly regular graphs with at most 40 vertices. In particular, we construct codes from adjacency or orbit matrices of graphs with parameters $(36, 15, 6, 6)$, $(36, 14, 4, 6)$, $(35, 16, 6, 8)$ and their complements, and from the graphs with parameters $(40, 12, 2, 4)$ and their complements. That completes the classification of self-orthogonal codes spanned by the adjacency matrices or orbit matrices of the strongly regular graphs with at most 40 vertices. Furthermore, we construct ternary codes of $2$-$(27,9,4)$ designs obtained as residual designs of the symmetric $(40, 13, 4)$ designs (complementary designs of the symmetric $(40, 27, 18)$ designs), and their ternary hulls. Some of the obtained codes are optimal, and some are best known for the given length and dimension.
Citation: Dean Crnković, Marija Maksimović, Bernardo Gabriel Rodrigues, Sanja Rukavina. Self-orthogonal codes from the strongly regular graphs on up to 40 vertices. Advances in Mathematics of Communications, 2016, 10 (3) : 555-582. doi: 10.3934/amc.2016026
##### References:
 [1] E. F. Assmus, Jr. and J. D. Key, Designs and their Codes, Cambridge Univ. Press, 1992. doi: 10.1017/CBO9781316529836. [2] 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. [3] T. Beth, D. Jungnickel and H. Lenz, Design Theory I, Cambridge Univ. Press, Cambridge, 1999. [4] 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. [5] I. Bouyukliev, On the binary projective codes with dimension 6, Discrete Appl. Math., 154 (2006), 1693-1708. doi: 10.1016/j.dam.2006.03.004. [6] I. Bouyukliev, V. Fack, W. Willems and J. Winne, Projective two-weight codes with small parameters and their corresponding graphs, Des. Codes Cryptogr., 41 (2006), 59-78. doi: 10.1007/s10623-006-0019-1. [7] A. E. Brouwer and W. H. Haemers, Structure and uniqueness of the $(81,20,1,6)$ strongly regular graph, Discrete Math., 106/107 (1992), 77-82. doi: 10.1016/0012-365X(92)90532-K. [8] D. Crnković, V. Mikulić Crnković and B. G. Rodrigues, Some optimal codes and strongly regular graphs from the linear group $L_4(3)$, Util. Math., 89 (2012), 237-255. [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. [10] D. Crnković and S. Rukavina, Construction of block designs admitting an abelian automorphism group, Metrika, 62 (2005), 175-183. doi: 10.1007/s00184-005-0407-y. [11] D. Crnković and S. Rukavina, On some symmetric $(45, 12, 3)$ and $(40,13, 4)$ designs, J. Comput. Math. Optim., 1 (2005), 55-63. [12] M. Grassl, Bounds on the minimum distance of linear codes and quantum codes, http://www.codetables.de, Accessed 9 June 2016. [13] 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:1008353723204. [14] N. Hamada, On the $p$-rank of the incidence matrix of a balanced or partially balanced incomplete block design and its application to error correcting codes, Hiroshima Math. J., 3 (1973), 153-226. [15] M. Harada and V. D. Tonchev, Self-orthogonal codes from symmetric designswith fixed-point-free automorphisms, Discrete Math., 264 (2003), 81-90. doi: 10.1016/S0012-365X(02)00553-8. [16] R. Hill and D. E. Newton, Optimal ternary linear codes, Des. Codes Cryptogr., 2 (1992), 137-157. doi: 10.1007/BF00124893. [17] Z. Janko, Coset enumeration in groups and constructions of symmetric designs, Ann. Discrete Math., 52 (1992), 275-277. doi: 10.1016/S0167-5060(08)70919-1. [18] C. Jansen, K. Lux, R. Parker and R. Wilson, An Atlas of Brauer Characters, Oxford Scient. Publ., Clarendon Press, 1995. [19] J. D. Key and K. Mackenzie-Fleming, Rigidity theorems for a class of affine resolvable designs, J. Combin. Math. Combin. Comput., 35 (2000), 147-160. [20] R. Mathon and A. Rosa, 2-$(v,k,\lambda)$ designs of small order, in Handbook of Combinatorial Designs (eds. C.J. Colbourn and J.H. Dinitz), Chapman and Hall/CRC, Boca Raton, 2007, 25-58. [21] B. D. McKay and E. Spence, Classification of regular two-graphs on 36 and 38 vertices, Austral. J. Combin., 24 (2001), 293-300. [22] B. G. Rodrigues, Self-orthogonal designs and codes from the symplectic groups $S_4(3)$ and $S_4(4)$, Discrete Math., 308 (2008), 1941-1950. doi: 10.1016/j.disc.2007.04.047. [23] B. G. Rodrigues, Some optimal codes related to graphs invariant under the alternating group $A_8$, Adv. Math. Commun., 5 (2011), 339-350. doi: 10.3934/amc.2011.5.339. [24] L. D. Rudolph, A class of majority logic decodable codes, IEEE Trans. Inform. Theory, 13 (1967), 305-307. [25] S. S. Sane and M. S. Shrikhande, Quasi-Symmetric Designs, Cambridge Univ. Press, 1991. doi: 10.1017/CBO9780511665615. [26] E. Spence, The strongly regular $(40,12,2,4)$ graphs, Electron. J. Combin., 7 (2000), \#22, pp. 4. [27] E. Spence, Strongly regular graphs on at most 64 vertices, http://www.maths.gla.ac.uk/ es/srgraphs.php, Accessed 9 June 2016. [28] V. D. Tonchev, Codes, in Handbook of Combinatorial Designs, (eds. C.J. Colbourn and J.H. Dinitz), Chapman and Hall/CRC, Boca Raton, 2007, 667-702.

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##### References:
 [1] E. F. Assmus, Jr. and J. D. Key, Designs and their Codes, Cambridge Univ. Press, 1992. doi: 10.1017/CBO9781316529836. [2] 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. [3] T. Beth, D. Jungnickel and H. Lenz, Design Theory I, Cambridge Univ. Press, Cambridge, 1999. [4] 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. [5] I. Bouyukliev, On the binary projective codes with dimension 6, Discrete Appl. Math., 154 (2006), 1693-1708. doi: 10.1016/j.dam.2006.03.004. [6] I. Bouyukliev, V. Fack, W. Willems and J. Winne, Projective two-weight codes with small parameters and their corresponding graphs, Des. Codes Cryptogr., 41 (2006), 59-78. doi: 10.1007/s10623-006-0019-1. [7] A. E. Brouwer and W. H. Haemers, Structure and uniqueness of the $(81,20,1,6)$ strongly regular graph, Discrete Math., 106/107 (1992), 77-82. doi: 10.1016/0012-365X(92)90532-K. [8] D. Crnković, V. Mikulić Crnković and B. G. Rodrigues, Some optimal codes and strongly regular graphs from the linear group $L_4(3)$, Util. Math., 89 (2012), 237-255. [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. [10] D. Crnković and S. Rukavina, Construction of block designs admitting an abelian automorphism group, Metrika, 62 (2005), 175-183. doi: 10.1007/s00184-005-0407-y. [11] D. Crnković and S. Rukavina, On some symmetric $(45, 12, 3)$ and $(40,13, 4)$ designs, J. Comput. Math. Optim., 1 (2005), 55-63. [12] M. Grassl, Bounds on the minimum distance of linear codes and quantum codes, http://www.codetables.de, Accessed 9 June 2016. [13] 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:1008353723204. [14] N. Hamada, On the $p$-rank of the incidence matrix of a balanced or partially balanced incomplete block design and its application to error correcting codes, Hiroshima Math. J., 3 (1973), 153-226. [15] M. Harada and V. D. Tonchev, Self-orthogonal codes from symmetric designswith fixed-point-free automorphisms, Discrete Math., 264 (2003), 81-90. doi: 10.1016/S0012-365X(02)00553-8. [16] R. Hill and D. E. Newton, Optimal ternary linear codes, Des. Codes Cryptogr., 2 (1992), 137-157. doi: 10.1007/BF00124893. [17] Z. Janko, Coset enumeration in groups and constructions of symmetric designs, Ann. Discrete Math., 52 (1992), 275-277. doi: 10.1016/S0167-5060(08)70919-1. [18] C. Jansen, K. Lux, R. Parker and R. Wilson, An Atlas of Brauer Characters, Oxford Scient. Publ., Clarendon Press, 1995. [19] J. D. Key and K. Mackenzie-Fleming, Rigidity theorems for a class of affine resolvable designs, J. Combin. Math. Combin. Comput., 35 (2000), 147-160. [20] R. Mathon and A. Rosa, 2-$(v,k,\lambda)$ designs of small order, in Handbook of Combinatorial Designs (eds. C.J. Colbourn and J.H. Dinitz), Chapman and Hall/CRC, Boca Raton, 2007, 25-58. [21] B. D. McKay and E. Spence, Classification of regular two-graphs on 36 and 38 vertices, Austral. J. Combin., 24 (2001), 293-300. [22] B. G. Rodrigues, Self-orthogonal designs and codes from the symplectic groups $S_4(3)$ and $S_4(4)$, Discrete Math., 308 (2008), 1941-1950. doi: 10.1016/j.disc.2007.04.047. [23] B. G. Rodrigues, Some optimal codes related to graphs invariant under the alternating group $A_8$, Adv. Math. Commun., 5 (2011), 339-350. doi: 10.3934/amc.2011.5.339. [24] L. D. Rudolph, A class of majority logic decodable codes, IEEE Trans. Inform. Theory, 13 (1967), 305-307. [25] S. S. Sane and M. S. Shrikhande, Quasi-Symmetric Designs, Cambridge Univ. Press, 1991. doi: 10.1017/CBO9780511665615. [26] E. Spence, The strongly regular $(40,12,2,4)$ graphs, Electron. J. Combin., 7 (2000), \#22, pp. 4. [27] E. Spence, Strongly regular graphs on at most 64 vertices, http://www.maths.gla.ac.uk/ es/srgraphs.php, Accessed 9 June 2016. [28] V. D. Tonchev, Codes, in Handbook of Combinatorial Designs, (eds. C.J. Colbourn and J.H. Dinitz), Chapman and Hall/CRC, Boca Raton, 2007, 667-702.
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