Group codes outperform binary-coset codes on nonbinary symmetric memoryless channels
IEEE Transactions on Information Theory
| Hi-index | 754.90 |
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-l), where V(n,d) = σi=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)≥c2n/V(n,d-1)log2V(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.