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arXiv · 0708.4164

Asymptotic improvement of the Gilbert-Varshamov bound for linear codes

Philippe Gaborit·Gilles Zemor
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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.

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Philippe Gaborit, Gilles Zemor. 2007-08-30. Asymptotic improvement of the Gilbert-Varshamov bound for linear codes. https://doi.org/10.1109/tit.2008.928288

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