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

TL;DR

This paper investigates the optimal rates for linear computation over multiple access channels using homologous codes, establishing bounds and showing the optimality of certain structured coding schemes.

Contribution

It derives inner and outer bounds on achievable rates with homologous codes and proves their optimality in matched scenarios, extending to broadcast channels.

Findings

Inner and outer bounds on rate tradeoffs are established.

Matching bounds imply existing coding schemes are optimal under certain conditions.

The techniques extend to characterizing broadcast channel rate regions.

Abstract

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.