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.  Google Scholar

[2]

T. Beth, D. Jungnickel and H. Lenz, "Design Theory I,'' Cambridge University Press, Cambridge, 1999.  Google Scholar

[3]

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

[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.  Google Scholar

[5]

I. Bouyukliev and J. Simonis, Some new results for optimal ternary linear codes, IEEE Trans. Inform. Theory, 48 (2002), 981-985.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

M. Grassl, Bounds on the minimum distance of linear codes and quantum codes,, available online at \url{http://www.codetables.de}, ().   Google Scholar

[11]

M. Grassl, Searching for linear codes with large minimum distance, in "Discovering Mathematics with Magma'' (eds. W. Bosma and J. Cannon), Springer, 2006.  Google Scholar

[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.  Google Scholar

[13]

Z. Janko, Coset enumeration in groups and constructions of symmetric designs, Ann. Discrete Math., 52 (1992), 275-277.  Google Scholar

[14]

S. Kageyama, A survey of resolvable solutions of balanced incomplete block designs, Rev. Inst. Internat. Statist., 40 (1972), 269-273. Google Scholar

[15]

P. Kaski and P. R. J. Östergård, There are exactly five biplanes with $k=11$, J. Combin. Des., 16 (2008), 117-127.  Google Scholar

[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.  Google Scholar

[17]

V. Krčadinac, "Steiner $2$-designs $S(k, 2k^2-2k+1)$,'' M.Sc thesis, University of Zagreb, 1999. Google Scholar

[18]

V. Krčadinac, "Construction and Classification of Finite Structures by Computer,'' Ph.D thesis, University of Zagreb, 2004. Google Scholar

[19]

V. Krčadinac, Some new Steiner $2$-designs $S(2,4,37)$, Ars Combin., 78 (2006), 127-135.  Google Scholar

[20]

R. A. Mathon, K. T. Phelps and A. Rosa, Small Steiner triple systems and their properties, Ars Combin., 15 (1983), 3-110.  Google Scholar

[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. Google Scholar

[22]

B. G. Rodrigues, Some codes related to the Gewirtz and Brouwer-Haemers graphs,, in preparation., ().   Google Scholar

[23]

S. Rukavina, Some new triplanes of order twelve, Glas. Mat. Ser. III, 36 (2001), 105-125.  Google Scholar

[24]

V. D. Tonchev, Codes, in "Handbook of Combinatorial Designs,'' Chapman and Hall/CRC, Boca Raton, (2007), 677-702.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

E. F. Assmus, Jr. and J. D. Key, "Designs and Their Codes,'' Cambridge University Press, Cambridge 1992.  Google Scholar

[2]

T. Beth, D. Jungnickel and H. Lenz, "Design Theory I,'' Cambridge University Press, Cambridge, 1999.  Google Scholar

[3]

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

[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.  Google Scholar

[5]

I. Bouyukliev and J. Simonis, Some new results for optimal ternary linear codes, IEEE Trans. Inform. Theory, 48 (2002), 981-985.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

M. Grassl, Bounds on the minimum distance of linear codes and quantum codes,, available online at \url{http://www.codetables.de}, ().   Google Scholar

[11]

M. Grassl, Searching for linear codes with large minimum distance, in "Discovering Mathematics with Magma'' (eds. W. Bosma and J. Cannon), Springer, 2006.  Google Scholar

[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.  Google Scholar

[13]

Z. Janko, Coset enumeration in groups and constructions of symmetric designs, Ann. Discrete Math., 52 (1992), 275-277.  Google Scholar

[14]

S. Kageyama, A survey of resolvable solutions of balanced incomplete block designs, Rev. Inst. Internat. Statist., 40 (1972), 269-273. Google Scholar

[15]

P. Kaski and P. R. J. Östergård, There are exactly five biplanes with $k=11$, J. Combin. Des., 16 (2008), 117-127.  Google Scholar

[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.  Google Scholar

[17]

V. Krčadinac, "Steiner $2$-designs $S(k, 2k^2-2k+1)$,'' M.Sc thesis, University of Zagreb, 1999. Google Scholar

[18]

V. Krčadinac, "Construction and Classification of Finite Structures by Computer,'' Ph.D thesis, University of Zagreb, 2004. Google Scholar

[19]

V. Krčadinac, Some new Steiner $2$-designs $S(2,4,37)$, Ars Combin., 78 (2006), 127-135.  Google Scholar

[20]

R. A. Mathon, K. T. Phelps and A. Rosa, Small Steiner triple systems and their properties, Ars Combin., 15 (1983), 3-110.  Google Scholar

[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. Google Scholar

[22]

B. G. Rodrigues, Some codes related to the Gewirtz and Brouwer-Haemers graphs,, in preparation., ().   Google Scholar

[23]

S. Rukavina, Some new triplanes of order twelve, Glas. Mat. Ser. III, 36 (2001), 105-125.  Google Scholar

[24]

V. D. Tonchev, Codes, in "Handbook of Combinatorial Designs,'' Chapman and Hall/CRC, Boca Raton, (2007), 677-702.  Google Scholar

[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.  Google Scholar

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