Relation between encoder and syndrome former variables and symbol reliability estimation using a syndrome trellis

We derive a linear correspondence between the variables of an encoder and those of a corresponding syndrome former. Using the derived correspondence, we show that the log-likelihood ratio of an information bit conditioned on a received sequence can be equally calculated using the syndrome trellis. It is shown that the proposed method also applies to recursive systematic convolutional codes which are typical constituent codes for turbo codes. Moreover, we show that soft-in syndrome decoding considering a priori probabilities of information bits is possible in the same way as for Viterbi decoding based on the code trellis. Hence, the proposed method can be applied to iterative decoding such as turbo decoding. We also show that the proposed method is effective for high-rate codes by making use of trellis modification.