Class of constructive asymptotically good algebraic codes

For any rate
R, 0 < R < 1
, 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.