Given positive integers n and d, let A2(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 A2(n,d)≥2n/V(n,d−1), where
V(n,d)=∑i=0d(in) is the volume of a Hamming sphere of
radius d. We show that, in fact, there exists a positive constant c such
that A2(n,d)≥cV(n,d−1)2nlog2V(n,d−1) whenever d/n≤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.Comment: 10 pages, 3 figures; to appear in the IEEE Transactions on
Information Theory, submitted August 12, 2003, revised March 28, 200