# 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, (1992). [2] T. Beth, D. Jungnickel and H. Lenz, "Design Theory I,'', Cambridge University Press, (1999). [3] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I: The user language,, J. Symbolic Comput., 24 (1997), 235. [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. 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. [6] A. E. Brouwer, Bounds on linear codes,, in, (1998), 295. [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. [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. [9] D. Crnković and S. Rukavina, Construction of block designs admitting an abelian automorphism group,, Metrika, 62 (2005), 175. doi: 10.1007/s00184-005-0407-y. [10] M. Grassl, Bounds on the minimum distance of linear codes and quantum codes,, available online at \url{http://www.codetables.de}, (). [11] M. Grassl, Searching for linear codes with large minimum distance,, in, (2006). [12] M. Harada and V. D. Tonchev, Self-orthogonal codes from symmetric designs with fixed-point-free automorphisms,, Discrete Math., 264 (2003), 81. 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. [14] S. Kageyama, A survey of resolvable solutions of balanced incomplete block designs,, Rev. Inst. Internat. Statist., 40 (1972), 269. [15] P. Kaski and P. R. J. Östergård, There are exactly five biplanes with $k=11$,, J. Combin. Des., 16 (2008), 117. [16] J. D. Key and V. D. Tonchev, Computational results for the known biplanes of order 9,, in, (1997), 113. [17] V. Krčadinac, "Steiner $2$-designs $S(k, 2k^2-2k+1)$,'', M.Sc thesis, (1999). [18] V. Krčadinac, "Construction and Classification of Finite Structures by Computer,'', Ph.D thesis, (2004). [19] V. Krčadinac, Some new Steiner $2$-designs $S(2,4,37)$,, Ars Combin., 78 (2006), 127. [20] R. A. Mathon, K. T. Phelps and A. Rosa, Small Steiner triple systems and their properties,, Ars Combin., 15 (1983), 3. [21] R. Mathon and A. Rosa, $2$-$(v,k, \lambda)$ designs of small order,, in, (2007), 25. [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. [24] V. D. Tonchev, Codes,, in, (2007), 677. [25] V. D. Tonchev and R. S. Weishaar, Steiner triple systems of order 15 and their codes,, J. Statist. Plann. Inference, 58 (1997), 207.

show all references

##### References:
 [1] E. F. Assmus, Jr. and J. D. Key, "Designs and Their Codes,'', Cambridge University Press, (1992). [2] T. Beth, D. Jungnickel and H. Lenz, "Design Theory I,'', Cambridge University Press, (1999). [3] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I: The user language,, J. Symbolic Comput., 24 (1997), 235. [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. 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. [6] A. E. Brouwer, Bounds on linear codes,, in, (1998), 295. [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. [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. [9] D. Crnković and S. Rukavina, Construction of block designs admitting an abelian automorphism group,, Metrika, 62 (2005), 175. doi: 10.1007/s00184-005-0407-y. [10] M. Grassl, Bounds on the minimum distance of linear codes and quantum codes,, available online at \url{http://www.codetables.de}, (). [11] M. Grassl, Searching for linear codes with large minimum distance,, in, (2006). [12] M. Harada and V. D. Tonchev, Self-orthogonal codes from symmetric designs with fixed-point-free automorphisms,, Discrete Math., 264 (2003), 81. 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. [14] S. Kageyama, A survey of resolvable solutions of balanced incomplete block designs,, Rev. Inst. Internat. Statist., 40 (1972), 269. [15] P. Kaski and P. R. J. Östergård, There are exactly five biplanes with $k=11$,, J. Combin. Des., 16 (2008), 117. [16] J. D. Key and V. D. Tonchev, Computational results for the known biplanes of order 9,, in, (1997), 113. [17] V. Krčadinac, "Steiner $2$-designs $S(k, 2k^2-2k+1)$,'', M.Sc thesis, (1999). [18] V. Krčadinac, "Construction and Classification of Finite Structures by Computer,'', Ph.D thesis, (2004). [19] V. Krčadinac, Some new Steiner $2$-designs $S(2,4,37)$,, Ars Combin., 78 (2006), 127. [20] R. A. Mathon, K. T. Phelps and A. Rosa, Small Steiner triple systems and their properties,, Ars Combin., 15 (1983), 3. [21] R. Mathon and A. Rosa, $2$-$(v,k, \lambda)$ designs of small order,, in, (2007), 25. [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. [24] V. D. Tonchev, Codes,, in, (2007), 677. [25] V. D. Tonchev and R. S. Weishaar, Steiner triple systems of order 15 and their codes,, J. Statist. Plann. Inference, 58 (1997), 207.
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