American Institute of Mathematical Sciences

May  2013, 7(2): 161-174. doi: 10.3934/amc.2013.7.161

Self-orthogonal codes from orbit matrices of 2-designs

 1 Department of Mathematics, University of Rijeka, Radmile Matejčić 2, 51000 Rijeka, Croatia, Croatia 2 School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban 4041, South Africa 3 Faculty of Engineering, University of Rijeka, Vukovarska 58, 51000 Rijeka, Croatia

Received  July 2012 Revised  November 2012 Published  May 2013

In this paper we present a method for constructing self-orthogonal codes from orbit matrices of $2$-designs that admit an automorphism group $G$ which acts with orbit lengths $1$ and $w$, where $w$ divides $|G|$. This is a generalization of an earlier method proposed by Tonchev for constructing self-orthogonal codes from orbit matrices of $2$-designs with a fixed-point-free automorphism of prime order. As an illustration of our method we provide a classification of self-orthogonal codes obtained from the non-fixed parts of the orbit matrices of the symmetric $2$-$(56,11,2)$ designs, some symmetric designs $2$-$(71,15,3)$ (and their residual designs), and some non-symmetric $2$-designs, namely those with parameters $2$-$(15,3,1)$, $2$-$(25,4,1)$, $2$-$(37,4,1)$, and $2$-$(45,5,1)$, respectively with automorphisms of order $p$, where $p$ is an odd prime. We establish that the codes with parameters $[10,4,6]_3$ and $[11,4,6]_3$ are optimal two-weight codes. Further, we construct an optimal binary self-orthogonal $[16,5,8]$ code from the non-fixed part of the orbit matrix of the $2$-$(64,8,1)$ design with respect to an automorphism group of order four.
Citation: Dean Crnković, Bernardo Gabriel Rodrigues, Sanja Rukavina, Loredana Simčić. Self-orthogonal codes from orbit matrices of 2-designs. Advances in Mathematics of Communications, 2013, 7 (2) : 161-174. doi: 10.3934/amc.2013.7.161
References:
 [1] E. F. Assmus, Jr. and J. D. Key, "Designs and Their Codes,'' Cambridge University Press, Cambridge 1992. [2] T. Beth, D. Jungnickel and H. Lenz, "Design Theory I,'' Cambridge University Press, Cambridge, 1999. [3] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I: The user language, J. Symbolic Comput., 24 (1997), 235-265. [4] 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. [5] I. Bouyukliev and J. Simonis, Some new results for optimal ternary linear codes, IEEE Trans. Inform. Theory, 48 (2002), 981-985. [6] A. E. Brouwer, Bounds on linear codes, in "Handbook of Coding Theory'' (eds. V.S. Pless and W.C. Huffman), Elsevier, (1998), 295-461. [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. [8] D. Crnković, B. G. Rodrigues, S. Rukavina and L. Simčić, Ternary codes from the strongly regular $(45,12,3,3)$ graphs and orbit matrices of $2$-$(45,12,3)$ designs, Discrete Math., 312 (2012), 3000-3010. [9] 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. [10] M. Grassl, Bounds on the minimum distance of linear codes and quantum codes, available online at http://www.codetables.de [11] M. Grassl, Searching for linear codes with large minimum distance, in "Discovering Mathematics with Magma'' (eds. W. Bosma and J. Cannon), Springer, 2006. [12] M. Harada and V. D. 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. [13] Z. Janko, Coset enumeration in groups and constructions of symmetric designs, Ann. Discrete Math., 52 (1992), 275-277. [14] S. Kageyama, A survey of resolvable solutions of balanced incomplete block designs, Rev. Inst. Internat. Statist., 40 (1972), 269-273. [15] P. Kaski and P. R. J. Östergård, There are exactly five biplanes with $k=11$, J. Combin. Des., 16 (2008), 117-127. [16] J. D. Key and V. D. Tonchev, Computational results for the known biplanes of order 9, in "Geometry, Combinatorial Designs and Related Structures'' (eds. J.W.P. Hirschfeld et al), Cambridge Univ. Press, Cambridge, (1997), 113-122. [17] V. Krčadinac, "Steiner $2$-designs $S(k, 2k^2-2k+1)$,'' M.Sc thesis, University of Zagreb, 1999. [18] V. Krčadinac, "Construction and Classification of Finite Structures by Computer,'' Ph.D thesis, University of Zagreb, 2004. [19] V. Krčadinac, Some new Steiner $2$-designs $S(2,4,37)$, Ars Combin., 78 (2006), 127-135. [20] R. A. Mathon, K. T. Phelps and A. Rosa, Small Steiner triple systems and their properties, Ars Combin., 15 (1983), 3-110. [21] R. Mathon and A. Rosa, $2$-$(v,k, \lambda)$ designs of small order, in "Handbook of Combinatorial Designs,'' Chapman and Hall/CRC, Boca Raton, (2007), 25-58. [22] B. G. Rodrigues, Some codes related to the Gewirtz and Brouwer-Haemers graphs, in preparation. [23] S. Rukavina, Some new triplanes of order twelve, Glas. Mat. Ser. III, 36 (2001), 105-125. [24] V. D. Tonchev, Codes, in "Handbook of Combinatorial Designs,'' Chapman and Hall/CRC, Boca Raton, (2007), 677-702. [25] V. D. Tonchev and R. S. Weishaar, Steiner triple systems of order 15 and their codes, J. Statist. Plann. Inference, 58 (1997), 207-216.

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References:
 [1] E. F. Assmus, Jr. and J. D. Key, "Designs and Their Codes,'' Cambridge University Press, Cambridge 1992. [2] T. Beth, D. Jungnickel and H. Lenz, "Design Theory I,'' Cambridge University Press, Cambridge, 1999. [3] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I: The user language, J. Symbolic Comput., 24 (1997), 235-265. [4] 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. [5] I. Bouyukliev and J. Simonis, Some new results for optimal ternary linear codes, IEEE Trans. Inform. Theory, 48 (2002), 981-985. [6] A. E. Brouwer, Bounds on linear codes, in "Handbook of Coding Theory'' (eds. V.S. Pless and W.C. Huffman), Elsevier, (1998), 295-461. [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. [8] D. Crnković, B. G. Rodrigues, S. Rukavina and L. Simčić, Ternary codes from the strongly regular $(45,12,3,3)$ graphs and orbit matrices of $2$-$(45,12,3)$ designs, Discrete Math., 312 (2012), 3000-3010. [9] 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. [10] M. Grassl, Bounds on the minimum distance of linear codes and quantum codes, available online at http://www.codetables.de [11] M. Grassl, Searching for linear codes with large minimum distance, in "Discovering Mathematics with Magma'' (eds. W. Bosma and J. Cannon), Springer, 2006. [12] M. Harada and V. D. 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. [13] Z. Janko, Coset enumeration in groups and constructions of symmetric designs, Ann. Discrete Math., 52 (1992), 275-277. [14] S. Kageyama, A survey of resolvable solutions of balanced incomplete block designs, Rev. Inst. Internat. Statist., 40 (1972), 269-273. [15] P. Kaski and P. R. J. Östergård, There are exactly five biplanes with $k=11$, J. Combin. Des., 16 (2008), 117-127. [16] J. D. Key and V. D. Tonchev, Computational results for the known biplanes of order 9, in "Geometry, Combinatorial Designs and Related Structures'' (eds. J.W.P. Hirschfeld et al), Cambridge Univ. Press, Cambridge, (1997), 113-122. [17] V. Krčadinac, "Steiner $2$-designs $S(k, 2k^2-2k+1)$,'' M.Sc thesis, University of Zagreb, 1999. [18] V. Krčadinac, "Construction and Classification of Finite Structures by Computer,'' Ph.D thesis, University of Zagreb, 2004. [19] V. Krčadinac, Some new Steiner $2$-designs $S(2,4,37)$, Ars Combin., 78 (2006), 127-135. [20] R. A. Mathon, K. T. Phelps and A. Rosa, Small Steiner triple systems and their properties, Ars Combin., 15 (1983), 3-110. [21] R. Mathon and A. Rosa, $2$-$(v,k, \lambda)$ designs of small order, in "Handbook of Combinatorial Designs,'' Chapman and Hall/CRC, Boca Raton, (2007), 25-58. [22] B. G. Rodrigues, Some codes related to the Gewirtz and Brouwer-Haemers graphs, in preparation. [23] S. Rukavina, Some new triplanes of order twelve, Glas. Mat. Ser. III, 36 (2001), 105-125. [24] V. D. Tonchev, Codes, in "Handbook of Combinatorial Designs,'' Chapman and Hall/CRC, Boca Raton, (2007), 677-702. [25] V. D. Tonchev and R. S. Weishaar, Steiner triple systems of order 15 and their codes, J. Statist. Plann. Inference, 58 (1997), 207-216.
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