Class of constructive asymptotically good algebraic codes

For any rate , a sequence of specific binary codes with rate and minimum distance 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 is the maximum of 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 With a simple algebraic description of the added digits. Alternatively, the codes are the concatenation of a Reed-Solomon outer code of length with 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 . This procedure can be carried out by a simple decoder which performs approximately computations.