A technique to prove the channel coding theorem for a reduced class of codes and its applications

The channel coding theorem for DMCs is proved using random coding. The standard proofs use the property of total randomness of the ensemble of codes. For a more restricted ensemble of codes a specific proof is needed. This has been done, for example, for linear codes, time-varying convolutional codes and for constant composition (constant type) codes. In this work we show that weaker demands on the ensembles of codes, than independency of the code words, are needed. As a result, we get simple proofs of the channel coding theorem for interesting restricted ensembles of codes. As an example we can prove the channel coding theorem for the ensembles of codes based on the output of an LFSR (linear feedback shift register). A particularly interesting application of our technique is a proof that fixed (time-invariant) convolutional codes attain capacity.