UC Riverside

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 rho(q, n, d) = n - log_(q) A(q, n, d) as n grows while q and d are fixed. For any d and q greater than or equal to d - 1, long algebraic codes are designed that improve on the Bose-Chaudhuri-Hocquenghem (BCH) codes and have the lowest asymptotic redundancy rho(q, n, d) less than or similar to ((d - 3) + 1/(d - 2)) log_(q)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.