Finite field matrix channels for network coding

TL;DR

This paper generalizes finite field matrix channels for network coding, allowing variable error counts, and shows that capacity can be achieved with a simple input distribution based on matrix rank.

Contribution

It extends previous models to variable error distributions and proves that capacity-achieving inputs depend only on matrix rank, simplifying analysis.

Findings

Capacity can be expressed as a maximization over rank distributions.

Capacity-achieving input distributions are uniform given the input matrix rank.

The set of possible input ranks is linear in size, simplifying optimization.

Abstract

In 2010, Silva, Kschischang and K\"otter studied certain classes of finite field matrix channels in order to model random linear network coding where exactly $t$ random errors are introduced. In this paper we consider a generalisation of these matrix channels where the number of errors is not required to be constant, indeed the number of errors may follow any distribution. We show that a capacity-achieving input distribution can always be taken to have a very restricted form (the distribution should be uniform given the rank of the input matrix). This result complements, and is inspired by, a paper of Nobrega, Silva and Uchoa-Filho, that establishes a similar result for a class of matrix channels that model network coding with link erasures. Our result shows that the capacity of our channels can be expressed as a maximisation over probability distributions on the set of possible ranks of input matrices: a set of linear rather than exponential size.