A Class of Non-Linearly Solvable Networks

TL;DR

This paper constructs specific networks that are solvable only over certain alphabet sizes, demonstrating limitations of linear solutions and capacity bounds in network coding.

Contribution

It introduces a class of networks with unique solvability properties depending on the alphabet size, especially highlighting non-linear solvability constraints.

Findings

Networks solvable only over specific alphabet sizes

No vector linear solution over any R-module alphabet for composite m

Linear capacity is bounded away from one for composite m

Abstract

For each integer $m \geq 2$, a network is constructed which is solvable over an alphabet of size $m$ but is not solvable over any smaller alphabets. If $m$ is composite, then the network has no vector linear solution over any $R$-module alphabet and is not asymptotically linear solvable over any finite-field alphabet. The network's capacity is shown to equal one, and when $m$ is composite, its linear capacity is shown to be bounded away from one for all finite-field alphabets.