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On the existence of Hadamard difference sets in groups of order 400
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 |
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. |
show all references
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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