Algebraic constructions of Shannon codes for regular channels

The problem of the explicit construction of encoders achieving Shannon's capacity and admitting a simple decoding algorithm is considered. A solution based on Justesen's idea of variable concatenated codes is given for the case of a symmetric memoryless channel with an input alphabet of prime power order, under the assumption that the information messages are equiprobable. This construction remains good for a nonsymmetric channel provided the encoding rate is smaller than a well-defined "pseudocapacity." In case the channel is regular, it is shown that the error probability after decoding is an exponentially decreasing function of the block length for any encoding rate less than the channel capacity.