Self-Dual Codes better than the Gilbert--Varshamov bound

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 self-dual codes better than the asymptotic Gilbert--Varshamov bound.