Bounds on information combining for parity-check equations

When several independent channels are coupled by a parity check constraint on their inputs, the mutual information between the input of one channel and the outputs of all other channels can be expressed as a combination of the mutual information between the input and the output of each individual channel. This concept is denoted as information combining. For binary-input symmetric discrete memoryless channels, we present bounds on the combined information which are only based on the mutual information of the channels. Furthermore, we show that these bounds cannot be further improved. Exact expressions are provided for the case that all channels are binary symmetric channels and for the case that all channels are binary erasure channels.