Improving the Gilbert-Varshamov bound for q-ary codes

Given positive integers q,n, and d, denote by A_q(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)≥qⁿ/V_q(n,d) where V_q(n,d)=Σ_i=0^d (_iⁿ)(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 α less than (q-1)/q there is a positive constant c such that for d≤αn A_q(n,d+1)≥cqⁿ/V_q(n,d)n. This confirms a conjecture by Jiang and Vardy.