Asymptotic improvement of the Gilbert-Varshamov bound for binary linear codes

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) = Sigma(d)(i=0) ((n)(i)) stands for the volume of a Hamming hall of radius d. Recently Jiang and Vardy showed that for binary non-linear codes this bound could be improved to A(2)(n, d) >= cn 2(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] double circulant codes satisfy the same improved Gilbert-Varshamov bound.