Abstract
In this paper, two general methods for constructing self-dual codes are presented. These methods use circulant matrices in circulant or bordered circulant structures to construct the suitable generator matrices. The necessary and sufficient conditions, for the generated codes to be self-dual, are provided. Special cases of the proposed methods include the well known “Pure Double Circulant” construction and the “Bordered Double circulant” construction of self-dual codes. As an example, the methods were applied to search for self-dual codes in GF(5). Many new inequivalent self-dual codes with best known distance are found.
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Betsumiya K, Georgiou S, Gulliver T.A., Harada M., Koukouvinos C.: On self-dual codes over some prime fields. Discr. Math. 262, 37–58 (2003)
Bosma W., Cannon J., Playoust C.: The Magma algebra system. I. The user language. J. Symbolic Comput. 24, 235–265 (1997)
Davis P.J.: Circulant Matrices. AMS Chelsea Publishing, New York (1994)
Gaborit P.: Tables of Self-Dual Codes. http://www.unilim.fr/pages_perso/philippe.gaborit/SD/index.html.
Gaborit P., Otmani A.: Experimental constructions of self-dual codes. Finite Fields Appl. 9, 372–394 (2003)
Grassl M., Gulliver T.A.: On circulant self-dual codes over small fields. Des. Codes Cryptogr. 52, 57–81 (2009)
Harada M., Gulliver T.A., Miyabayashi H.: Double circulant and quasi-twisted self-dual codes over \({\mathbb{F}_5}\) and \({\mathbb{F}_7}\). Adv. Math. Commun. 1, 223–238 (2007)
MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1977)
Rains E., Sloane N.J.A.: Self-dual codes. In: Pless, V. et al. (eds) Handbook of Coding Theory, Elsevier, Amsterdam (1998)
Tonchev V.D.: Codes, chapter VII.1. In: Colbourn, C.J., Dinitz, J.H. (eds) Handbook of Combinatorial Designs, 2nd edn, pp. 677–702. Chapman & Hall/CRC, Boca Raton (2007)
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This is one of several papers published together in Designs, Codes and Cryptography on the special topic: “Geometry, Combinatorial Designs & Cryptology”.
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Georgiou, S.D., Lappas, E. Self-dual codes from circulant matrices. Des. Codes Cryptogr. 64, 129–141 (2012). https://doi.org/10.1007/s10623-011-9510-4
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DOI: https://doi.org/10.1007/s10623-011-9510-4