Des. Codes Cryptogr. https://doi.org/10.1007/s10623-018-0497-y 1 2 Alp Bassa · Henning Stichtenoth Received: 22 September 2017 / Revised: 8 May 2018 / Accepted: 18 May 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018 Abstract We show that every self-orthogonal code over F 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 F , with q ≥ 64, except possibly q = 125, there are inﬁnite families of self-dual codes, which are asymptotically better than the asymptotic Gilbert–Varshamov bound. Keywords Self-dual codes · Algebraic geometry codes · Gilbert–Varshamov Bound · Tsfasman–Vladut–Zink Bound · Towers of function ﬁelds · Asymptotically good codes · Quadratic forms · Witt’s Theorem Mathematics Subject Classiﬁcation 14G50 · 94B27 · 94B65 · 15A63 · 11T71 1 Introduction Let F be the ﬁnite ﬁeld of cardinality q where q is a power of some prime number p.Wemean by a code C over F always a linear code; i.e., C is a linear subspace of the n-dimensional Communicated by J. Bierbrauer. A.B. was supported by the BAGEP Award of the
Designs, Codes and Cryptography – Springer Journals
Published: May 29, 2018
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