Improving the Gilbert-Varshamov bound for q-ary codes

Given positive integers q,n, and d, denote by A/sub 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/sub q/(n,d+1)/spl ges/q/sup n//V/sub q/(n,d) where V/sub q/(n,d)=/spl Sigma//sub i=0//sup d/ (/sub i//sup n/)(q-1)/sup 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 /spl alpha/ less than (q-1)/q there is a positive constant c such that for d/spl les//spl alpha/n A/sub q/(n,d+1)/spl ges/cq/sup n//V/sub q/(n,d)n. This confirms a conjecture by Jiang and Vardy.