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

TL;DR

This paper demonstrates that for large nonprime fields, there exist self-dual codes surpassing the Gilbert--Varshamov bound asymptotically, using algebraic function field towers.

Contribution

It establishes the existence of self-dual codes exceeding the Gilbert--Varshamov bound over large nonprime fields, extending previous bounds with algebraic geometry methods.

Findings

Self-dual codes exist beyond the Gilbert--Varshamov bound for q ≥ 64.

Extension of self-orthogonal codes to self-dual codes under certain conditions.

Use of Galois towers of algebraic function fields to construct better codes.

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 $q\geq 64$, except possibly $q=125$, there are self-dual codes better than the asymptotic Gilbert--Varshamov bound.