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)ges2
ⁿ
/V(n,d-1) where V(n,d)=XiEd
_i=0
(En
_i
) stands for the volume of a Hamming ball of radius d. Recently Jiang and Vardy showed that for binary non-linear codes this bound could be improved to A
₂
(n,d)gescn2
ⁿ
/V(n,d -1) for c a constant and d/nles0.499. 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