A generalisation of the Gilbert-Varshamov bound and its asymptotic evaluation

The Gilbert-Varshamov (GV) lower bound on the maximum cardinality of a q-ary code of length n with minimum Hamming distance at least d can be obtained by application of Turán’s theorem to the graph with vertex set {0, 1, . . . , q − 1}n in which two vertices are joined if and only if their Hamming distance is at least d. We generalize the GV bound by applying Turán’s theorem to the graph with vertex set Cn, where C is a q-ary code of length m and two vertices are joined if and only if their Hamming distance at least d. We asymptotically evaluate the resulting bound for n → ∞ and d ∼ δmn for fixed δ > 0, and derive conditions on the distance distribution of C that are necessary and sufficient for the asymptotic generalized bound to beat the asymptotic GV bound. By invoking the Delsarte inequalities, we conclude that no improvement on the asymptotic GV bound is obtained. By using a sharpening of Turán’s theorem due to Caro and Wei, we improve on our bound. It is undecided if there exists a code C for which the improved bound can beat the asymptotic GV bound.