Asymptotic improvement of the Gilbert-Varshamov bound for linear codes

Philippe Gaborit (1) , Gilles Zemor (2)

1 DMI
2 IMB - Institut de Mathématiques de Bordeaux

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.

Domaines: Théorie de l'information et codage [math.IT]; Théorie de l'information [cs.IT]

Philippe Gaborit, Gilles Zemor. Asymptotic improvement of the Gilbert-Varshamov bound for linear codes. IEEE Transactions on Information Theory, 2008, 54 (9), pp.3865-3872. ⟨10.1109/TIT.2008.928288⟩. ⟨hal-00181471⟩