Some results on communicating the sum of sources over a network

TL;DR

This paper explores the conditions under which the sum of multiple sources can be communicated to terminals in a network, revealing that the ability depends on the network's structure and the characteristic of the finite field used.

Contribution

It demonstrates that for any finite set of primes, there exist networks where sum communication is only possible over finite fields with characteristics in that set, highlighting field characteristic constraints.

Findings

Sum communication depends on network connectivity and field characteristic.

Existence of networks where sum can only be communicated over specific finite fields.

Some networks cannot communicate the sum over any finite field despite full connectivity.

Abstract

We consider the problem of communicating the sum of $m$ sources to $n$ terminals in a directed acyclic network. Recently, it was shown that for a network of unit capacity links with either $m=2$ or $n=2$, the sum of the sources can be communicated to the terminals if and only if every source-terminal pair is connected in the network. We show in this paper that for any finite set of primes, there exists a network where the sum of the sources can be communicated to the terminals only over finite fields of characteristic belonging to that set. As a corollary, this gives networks where the sum can not be communicated over any finite field even though every source is connected to every terminal.