Generalized Fano and non-Fano networks

TL;DR

This paper generalizes Fano and non-Fano networks to create networks with vector linear solutions that depend on arbitrary sets of prime characteristics, expanding understanding of linear network coding limitations.

Contribution

It introduces new network constructions that extend Fano and non-Fano properties to arbitrary prime sets, demonstrating broader conditions for linear solvability.

Findings

Networks can be designed with solutions dependent on specific prime sets.

Linear solutions exist for all vector dimensions if and only if the field characteristic matches the set.

The constructions generalize previous Fano and non-Fano network properties.

Abstract

It is known that the Fano network has a vector linear solution if and only if the characteristic of the finite field is $2$; and the non-Fano network has a vector linear solution if and only if the characteristic of the finite field is not $2$. Using these properties of Fano and non-Fano networks it has been shown that linear network coding is insufficient. In this paper we generalize the properties of Fano and non-Fano networks. Specifically, by adding more nodes and edges to the Fano network, we construct a network which has a vector linear solution for any vector dimension if and only if the characteristic of the finite field belongs to an arbitrary given set of primes $\{p_1,p_2,\ldots,p_l\}$. Similarly, by adding more nodes and edges to the non-Fano network, we construct a network which has a vector linear solution for any vector dimension if and only if the characteristic of the finite field does not belong to an arbitrary given set of primes $\{p_1,p_2,\ldots,p_l\}$.