Asymptotic improvement of the Gilbert-Varshamov bound for linear codes

TL;DR

This paper demonstrates that specific families of linear binary codes can asymptotically surpass the classical Gilbert-Varshamov bound, previously known mainly for non-linear codes, by using random double circulant codes.

Contribution

It shows that certain asymptotic families of linear codes, specifically random double circulant codes, achieve improved bounds similar to non-linear codes.

Findings

Linear codes can asymptotically improve the Gilbert-Varshamov bound.

Random double circulant codes meet the improved bound.

The result extends the bound's applicability to linear code families.

Abstract

The Gilbert-Varshamov bound states that the maximum size A_2(n,d) of a binary code of length n and minimum distance d satisfies A_2(n,d) >= 2^n/V(n,d-1) where V(n,d) stands for the volume of a Hamming ball of radius d. Recently Jiang and Vardy showed that for binary non-linear codes this bound can be improved to A_2(n,d) >= cn2^n/V(n,d-1) for c a constant and d/n <= 0.499. In this paper we show that certain asymptotic families of linear binary [n,n/2] random double circulant codes satisfy the same improved Gilbert-Varshamov bound.