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 vi and minimum distance d. The famous Gilbert-Varshamov bound asserts that A_q(n,d+1) ≥ qⁿ/V_q(n,d) where 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. This confirms a conjecture by Jiang and Vardy. [ABSTRACT FROM AUTHOR]