Bit Error Probability and Encoder Structure

The problem of decoding a binary linear code used over a binary symmetric channel so as to minimize bit error probability, rather than codeword error probability, is studied. In general, there are substantial differences with the maximum likelihood case. The optimal decoding rule depends on the crossover probability of the channel, the bit error probability depends on the generator matrix used, and the optimal generator matrix itself depends on the crossover probability of the channel. This paper demonstrates each of the above properties. In addition, an infinite class of codes whose optimal generator matrix is non-systematic when the channel crossover probability is small is demonstrated. A new demonstration of the following facts is presented: if the code has d? > 2, then for a sufficiently noisy channel, the optimal generator matrix is (uniquely) a (widesense) systematic one, and the optimal decoding rule is to ignore the parity checks and to take the information bits as they are received. The infinite class mentioned above has d? > 2, and thus has different optimal generator matrices at different channel crossover probabilities.