On characteristics dependent linear coding capacity of sum‐networks

Abstract

It is known that the linear coding capacity of non-multicast networks may depend on the characteristic of the finite field. Such a dependence has been demonstrated in several examples presented in the literature. However, in all such examples, the number of distinct values that the linear coding capacity can take over different characteristics of the finite field is two. For example, the Fano network has linear coding capacity equal to one over even characteristic, and has linear coding capacity equal to 4/5 over odd characteristics. It is a natural question that whether a network can have arbitrarily many different characteristics dependent linear coding capacities. In this study, the authors answer the question for sum-networks, which are networks where the terminals demand the sum of the messages generated by the sources. Specifically, they show that given a set of characteristics (prime numbers) {p 1 , p 2 , …, p n }, indeed there exists a sum-network such that for each different value of the characteristic of the finite field, the linear coding capacity of the sum-network is different.