Asymptotic improvement of the Gilbert-Varshamov bound on the size of binary codes

Given positive integers n and d, let A(2) (n, d) denote the maximum size of a binary code of length n and minimum distance d. The well-known Gilbert-Varshamov bound asserts that A(2) (n, d) greater than or equal to 2(n)/V (n, d - 1), where V (n, d) = Sigma(i=0)(d) (7) is the volume of a Hamming sphere of radius d. We show that, in fact, there exists a positive constant c such that A(2) (n, d) greater than or equal to c 2(n) / V(n, d-1) log(2) V(n, d-1) whenever d/n less than or equal to, 0.499. The result follows by recasting the Gilbert-Varshamov bound into a graph-theoretic framework and using the fact that the corresponding graph is locally sparse. Generalizations and extensions of this result are briefly discussed.