On the Optimal Achievable Rates for Linear Computation With Random Homologous Codes

The problem of computing a linear combination of sources over a multiple access channel is studied. Inner and outer bounds on the optimal tradeoff between the communication rates are established when encoding is restricted to random ensembles of homologous codes, namely, structured nested coset codes from the same generator matrix and individual shaping functions, but when decoding is optimized with respect to the realization of the encoders. For the special case in which the desired linear combination is “matched” to the structure of the multiple access channel in a natural sense, these inner and outer bounds coincide. This result indicates that most, if not all, coding schemes for computation in the literature that rely on random construction of nested coset codes cannot be improved by using more powerful decoders such as the maximum likelihood decoder. The proof techniques are adapted to characterize the rate region for broadcast channels achieved by Marton's (random) coding scheme under maximum likelihood decoding. By generalizing some of the techniques, a single-letter outer bound for the capacity region of the computation problem is presented and compared with the inner bound achieved by homologous codes.