On Modulo-Sum Computation Over an Erasure Multiple-Access Channel

Abstract

We study modulo-sum computation of two binary source sequences over a two-user erasure multiple access channel. The channel is modeled as a binary-input, erasure multiple access channel, which can be in one of three states-either the channel output is a modulo-sum of the two input symbols, or the channel output equals the input symbol on the first link and an erasure on the second link, or vice versa. The associated state sequence is independent and identically distributed. Unlike previously studied multiple-access channels, the proposed channel is not matched to the modulo-sum function and therefore we expect simple cut-set upper bounds to be far from capacity. In this paper, we establish a new upper bound on the modulo-sum capacity that is tighter than the cut-set bound. The key step in establishing this new bound is to provide suitable side information to the encoders to reduce the setup to a compound multiple-access channel and then capture the tension across multiple receivers required to compute the modulo-sum function. In our lower bound, it suffices to use identical linear codebooks at the two encoders. When a (strictly) causal feedback of the channel state is available to the encoders, we present a simple coding scheme that can achieve a rate larger than our upper bound for the case without feedback. This shows that the modulo-sum capacity is increased with feedback. An extension to the case of lossy reconstruction is also treated briefly.