Codebook Mismatch can be Fully Compensated by Mismatched Decoding

We consider an ensemble of constant composition codes that are subsets of linear codes: while the encoder uses only the constant-composition subcode, the decoder operates as if the full linear code was used, with the motivation of simultaneously benefiting both from the probabilistic shaping of the channel input (to achieve higher rates) and from the linear structure of the code (to allow for lower complexity practical decoding). We prove that the codebook mismatch can be fully compensated by using a mismatched additive decoding metric that achieves the random coding error exponent of (non-linear) constant composition codes. As the coding rate tends to the mutual information, the optimal mismatched metric approaches the maximum a posteriori probability (MAP) metric, showing that codebook mismatch with MAP metric is capacity-achieving for the optimal input assignment.