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Class of constructive asymptotically good algebraic codes

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For any rateR, 0 < R < 1, a sequence of specific(n,k)binary codes with rateR_n > Rand minimum distancedis 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), whereris the maximum offrac{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 overGF(2^m)With a simple algebraic description of the added digits. Alternatively, the codes are the concatenation of a Reed-Solomon outer code of lengthN = 2^m - 1withNdistinct 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 ond. This procedure can be carried out by a simple decoder which performs approximatelyn^2 log ncomputations.

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Information Theory, IEEE Transactions on  (Volume:18 ,  Issue: 5 )