Long Nonbinary Codes Exceeding the Gilbert - Varshamov Bound for any Fixed Distance
Abstract
Let A(q,n,d) denote the maximum size of a q-ary code of length n and distance d. We study the minimum asymptotic redundancy (q,n,d)=n-logq A(q,n,d) as n grows while q and d are fixed. For any d and q<=d-1, long algebraic codes are designed that improve on the BCH codes and have the lowest asymptotic redundancy (q,n,d) <= ((d-3)+1/(d-2)) logq n known to date. Prior to this work, codes of fixed distance that asymptotically surpass BCH codes and the Gilbert-Varshamov bound were designed only for distances 4,5 and 6.
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