Improving the Gilbert-Varshamov bound for q-ary codes
Given positive integers q, n, and d, denote by A, (n, d) the maximum size of a q-ary code of length n and minimum distance d. The famous Gilbert-Varshamov bound asserts that A(q)(n, d + 1) >= q(n)/V-q (n, d) where V-q (n, d) = Sigma(i=0)(d) ((n)(i)) (q - 1)(i) is the volume of a q-ary sphere of radius d. Extending a recent work of Jiang and Vardy on binary codes, we show that for any positive constant a less than (q - 1)/q there is a positive constant c such that for d  = cq(n)/V-q(n,d)n. This confirms a conjecture by Jiang and Vardy.
gilbert-varshamov bound q-ary codes entropy function independence number locally sparse graphs polynomial equivalence