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Designs, Codes and Cryptography

Erschienen in:

29.05.2018

Self-dual codes better than the Gilbert–Varshamov bound

verfasst von: Alp Bassa, Henning Stichtenoth

Erschienen in: Designs, Codes and Cryptography | Ausgabe 1/2019

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Abstract

We show that every self-orthogonal code over \({\mathbb {F}}_q\) of length n can be extended to a self-dual code, if there exists self-dual codes of length n. Using a family of Galois towers of algebraic function fields we show that over any nonprime field \({\mathbb {F}}_q\), with \(q\ge 64\), except possibly \(q=125\), there are infinite families of self-dual codes, which are asymptotically better than the asymptotic Gilbert–Varshamov bound.

Vorheriger Artikel On the q-bentness of Boolean functions

Bassa A., Beelen P., Garcia A., Stichtenoth H.: Galois towers over non-prime finite fields. Acta Arith. 164, 163–179 (2014).MathSciNetCrossRefMATH

MacWilliams F.J., Sloane N.J.A., Thompson J.G.: Good self-dual codes exist. Discret. Math. 3, 153–162 (1972).MathSciNetCrossRefMATH

Pless V., Pierce J.N.: Self-dual codes over GF(q) satisfy a modied Varshamov–Gilbert bound. Inf. Control 23, 35–40 (1973).CrossRefMATH

Pless V.: On Witt’s theorem for nonalternating symmetric bilinear forms over a field of characteristic 2. Proc. Am. Math. Soc. 15, 979–983 (1964).MathSciNetMATH

Pless V.: On the uniqueness of the Golay codes. J. Comb. Theory 5, 215–228 (1968).MathSciNetCrossRefMATH

Serre J.-P.: A Course in Arithmetic. Springer Verlag, New York-Heidelberg (1973).CrossRefMATH

Stichtenoth H.: Self-dual Goppa codes. J. Pure Appl. Algebra 55(1–2), 199–211 (1988).MathSciNetCrossRefMATH

Stichtenoth H.: Algebraic Function Fields and Codes, Graduate Texts in Mathematics 254. Springer-Verlag, Berlin (2009).

Stichtenoth H.: Transitive and self-dual codes attaining the Tsfasman-Vlăduţ-Zink bound. IEEE Trans. Inform. Theory 52(5), 2218–2224 (2006).MathSciNetCrossRefMATH

Titel: Self-dual codes better than the Gilbert–Varshamov bound
verfasst von: Alp Bassa
Henning Stichtenoth
Publikationsdatum: 29.05.2018
Verlag: Springer US
Erschienen in: Designs, Codes and Cryptography / Ausgabe 1/2019
Print ISSN: 0925-1022
Elektronische ISSN: 1573-7586
DOI: https://doi.org/10.1007/s10623-018-0497-y

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