Feasible alphabets for communicating the sum of sources over a network

TL;DR

This paper investigates the conditions under which sum-networks can be solved over different finite fields, revealing that solvability depends on the field's characteristic and establishing a duality with reverse networks.

Contribution

It constructs sum-networks with solvability constraints based on the field's characteristic and proves the equivalence of solvability between a network and its reverse.

Findings

Sum-networks solvable only over fields with certain prime characteristics.

Existence of a sum-network solvable over all fields except F_2.

Solvability of a sum-network is equivalent to that of its reverse network.

Abstract

We consider directed acyclic {\em sum-networks} with $m$ sources and $n$ terminals where the sources generate symbols from an arbitrary alphabet field $F$, and the terminals need to recover the sum of the sources over $F$. We show that for any co-finite set of primes, there is a sum-network which is solvable only over fields of characteristics belonging to that set. We further construct a sum-network where a scalar solution exists over all fields other than the binary field $F_2$. We also show that a sum-network is solvable over a field if and only if its reverse network is solvable over the same field.