Class of constructive asymptotically good algebraic codes

Abstract

For any rate R, 0 , a sequence of specific (n,k) binary codes with rate R_n > R and minimum distance d is constructed such that \begin{equation} \lim_{n \rightarrow \infty} \inf \frac{d}{n} \geq (1 - r ^{-1} R)H^{-1} (1 - r)> 0 \end{equation} (and hence the codes are asymptotically good), where r is the maximum of \frac{1}{2} and the solution of \begin{equation} R = \frac{r^2}{1 + \log_2 [1 - H^{-1}(1 - r)]}. \end{equation} The codes are extensions of the Reed-Solomon codes over GF(2^m) With a simple algebraic description of the added digits. Alternatively, the codes are the concatenation of a Reed-Solomon outer code of length N = 2^m - 1 with N distinct inner codes, namely all the codes in Wozeneraft's ensemble of randomly shifted codes. A decoding procedure is given that corrects all errors guaranteed correctable by the asymptotic lower bound on d . This procedure can be carried out by a simple decoder which performs approximately n^2 \log n computations.