Sum-networks: Dependency on Characteristic of the Finite Field under Linear Network Coding

TL;DR

This paper constructs sum-networks with capacities matching any positive rational number, where the existence of linear solutions depends on the characteristic of the finite field, extending previous results on prime sets.

Contribution

It demonstrates the existence of sum-networks with specified capacities and characteristic-dependent solvability, generalizing prior work to arbitrary rational capacities.

Findings

Sum-networks can be constructed for any rational capacity.

Existence of solutions depends on the finite field's characteristic.

Constructed networks match capacity and characteristic constraints.

Abstract

Sum-networks are networks where all the terminals demand the sum of the symbols generated at the sources. It has been shown that for any finite set/co-finite set of prime numbers, there exists a sum-network which has a vector linear solution if and only if the characteristic of the finite field belongs to the given set. It has also been shown that for any positive rational number $k/n$, there exists a sum-network which has capacity equal to $k/n$. It is a natural question whether, for any positive rational number $k/n$, and for any finite set/co-finite set of primes $\{p_1,p_2,\ldots,p_l\}$, there exists a sum-network which has a capacity achieving rate $k/n$ fractional linear network coding solution if and only if the characteristic of the finite field belongs to the given set. We show that indeed there exists such a sum-network by constructing such a sum-network.