Asymptotic Improvement of the Gilbert–Varshamov Bound for Linear Codes

The Gilbert-Varshamov (GV) bound states that the maximum size A<sub>2</sub>(n, d) of a binary code of length n and minimum distance d satisfies A<sub>2</sub>(n, d)ges2<sup>n</sup>/V(n, d-1) where V(n, d)=Sigma<sub>i=0</sub> <sup>d</sup>(<sub>i</sub> <sup>n</sup>) stands for the volume of a Hamming ball of radius d. Recently, Jiang and Vardy showed that for binary nonlinear codes this bound can be improved to A<sub>2</sub>(n, d)gescn2<sup>n</sup>/(V(n, d-1)) for c a constant and d/nges0.499. In this paper, we show that certain asymptotic families of linear binary [n, n/2] random double circulant codes satisfy the same improved GV bound.