Block and trellis codes for the binary (1-D) partial response channel with simple maximum likelihood decoders

We consider block and convolutional codes for improving the reliability of data transmission over the binary precoded noisy (1-D) partial response channel. We concentrate on a class of codes for which the maximum likelihood decoder, matched to the encoder, precoder, and the channel has the same trellis structure as the encoder. Thus, doubling the number of states due to the channel memory is avoided, We show that the necessary and sufficient condition to belong to this class is that all codewords be of the same parity. The Reed-Muller and Golay codes belong to this class