Asymptotic improvement of the Gilbert-Varshamov bound for linear codes
The Gilbert-Varshamov (GV) 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 ball of radius d. Recently, Jiang and Vardy showed that for binary nonlinear codes this bound can 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] random double circulant codes satisfy the same improved GV bound.
linear code gilbert-varshamov (gv) bound double circulant code random coding