Abstract
Combinatorial designs have been widely used, in the construction of self-dual codes. Recently, new methods of constructing self-dual codes are established using orthogonal designs (ODs), generalized orthogonal designs (GODs), a set of four sequences and Diophantine equations over GF(p). These methods had led to the construction of many new self-dual codes over small finite fields and rings. In this paper, we used some methods to construct self-orthogonal and self dual codes over GF(p), for some primes p. The construction is achieved by using some special kinds of combinatorial designs like orthogonal designs and GODs. Moreover, we combine eight circulant matrices, a system of Diophantine equations over GF(p), and a recently discovered array to obtain a new construction method. Using this method new self-dual and self-orthogonal codes are obtained. Specifically, we obtain new self-dual codes [32,16,12] over GF(11) and GF(13) which improve the previously known distances.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
K. T. Arasu and T. A. Gulliver, Self-dual codes over Fp and weighing matrices, IEEE Trans. Inform. Theory, Vol. 47 (2001) pp. 2051–2055.
K. Betsumiya, S. Georgiou, T. A. Gulliver, M. Harada and C. Koukouvinos, On self-dual codes over some prime fields, Discrete Mathematics, Vol. 262 (2003) pp. 37–58.
S. Georgiou and C. Koukouvinos, New self-dual codes over GF(5). In M. Walker (ed.), Cryptography and Coding, Lecture Notes in Computer Science, Vol. 1746, Springer-Verlag, Heidelberg (1999) pp. 63–69.
S. Georgiou and C. Koukouvinos, Self-dual codes over GF(7) and orthogonal designs, Utilitas Math., Vol. 60 (2001) pp. 79–89.
S. Georgiou and C. Koukouvinos, MDS self-dual codes over large prime fields, Finite Fields and Their Applications, Vol. 8 (2002) pp. 455–470.
S. Georgiou, M. Harada and C. Koukouvinos, Orthogonal designs and type II codes over Z2k, Designs, Codes and Cryptography, Vol. 25 (2002) pp. 163–174.
S. Georgiou, C. Koukouvinos and J. Seberry, Generalized orthogonal designs, Ars Combinatoria (to appear).
S. Georgiou and C. Koukouvinos, Self-dual codes over Fp and orthogonal designs, J. Combin. Math. Combin. Comput. (to appear).
A. V. Geramita and J. Seberry, Orthogonal designs: Quadratic forms and Hadamard matrices, Marcel Dekker, New York, Basel (1979).
T. A. Gulliver and M. Harada, New optimal self-dual codes over GF(7), Graphs and Combin., Vol. 15 (1999) pp. 175–186.
R. Hill, A First Course in Coding Theory, Clarendon Press, Oxford (1985).
J. S. Leon, V. Pless and N. J. A. Sloane, Self-dual codes over GF(5), J. Combin. Theory Ser. A, Vol. 32 (1982) pp. 178–194.
F. J. MacWilliams, C. L. Mallows and N. J. A. Sloane, Generalizations of Gleason's theorem on weight enumerators of self-dual codes, IEEE Trans. Inform. Theory, Vol. 18 (1972) pp. 794–805.
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland, Amsterdam (1977).
H. Kharaghani, Arrays for orthogonal designs, J. Combin. Designs, Vol. 8 (2000) pp. 166–173.
V. S. Pless and V. D. Tonchev, Self-dual codes over GF(7), IEEE Trans. Inform. Theory, Vol. 33 (1987) pp. 723–727.
E. M. Rains and N. J. A. Sloane, Self-dual codes. In V. Pless and W. C. Huffman (eds.), Handbook of Coding Theory, Amsterdam, Elsevier (1998) pp. 177–294.
M. Ventou and C. Rigoni, Self-dual doubly circulant codes, Discrete Math., Vol. 56 (1985) pp. 291–298.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Georgiou, S., Koukouvinos, C. Self-Orthogonal and Self-Dual Codes Constructed via Combinatorial Designs and Diophantine Equations. Designs, Codes and Cryptography 32, 193–206 (2004). https://doi.org/10.1023/B:DESI.0000029222.35938.bf
Issue Date:
DOI: https://doi.org/10.1023/B:DESI.0000029222.35938.bf