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Improving the Gilbert-Varshamov bound for q-ary codes

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2 Author(s)
Vu, V. ; Dept. of Math., Univ. of California, La Jolla, CA, USA ; Wu, L.

Given positive integers q,n, and d, denote by Aq(n,d) the maximum size of a q-ary code of length n and minimum distance d. The famous Gilbert-Varshamov bound asserts that Aq(n,d+1)≥qn/Vq(n,d) where Vq(n,d)=Σi=0d (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 Aq(n,d+1)≥cqn/Vq(n,d)n. This confirms a conjecture by Jiang and Vardy.

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Information Theory, IEEE Transactions on  (Volume:51 ,  Issue: 9 )