A generalisation of the Gilbert-Varshamov bound and its asymptotic evaluation

TL;DR

This paper generalizes the Gilbert-Varshamov bound for q-ary codes using Turan's theorem, evaluates its asymptotic behavior, and explores conditions under which it can outperform the classical bound, with improvements possible via a refined Turan's theorem.

Contribution

The paper introduces a generalized GV bound applying Turan's theorem to codes, evaluates its asymptotic limits, and identifies conditions for potential improvements over the classical bound.

Findings

Asymptotic evaluation of the generalized bound for large n

Conditions on code distance distribution for bound improvement

Use of a sharpened Turan's theorem to enhance the bound

Abstract

The Gilbert-Varshamov (GV) lower bound on the maximum cardinality of a q-ary code of length n with minimum Hamming distance at least d can be obtained by application of Turan's theorem to the graph with vertex set {0,1,..,q-1}^n in which two vertices are joined if and only if their Hamming distance is at least d. We generalize the GV bound by applying Turan's theorem to the graph with vertex set C^n, where C is a q-ary code of length m and two vertices are joined if and only if their Hamming distance at least d. We asymptotically evaluate the resulting bound for n-> \infty and d \delta mn for fixed \delta > 0, and derive conditions on the distance distribution of C that are necessary and sufficient for the asymptotic generalized bound to beat the asymptotic GV bound. By invoking the Delsarte inequalities, we conclude that no improvement on the asymptotic GV bound is obtained. By using a sharpening of Turan's theorem due to Caro and Wei, we improve on our bound. It is undecided if there exists a code C for which the improved bound can beat the asymptotic GV bound.