The capacity of a finite field matrix channel

TL;DR

This paper generalizes the capacity analysis of the Additive-Multiplicative Matrix Channel over finite fields, removing previous size constraints and refining capacity estimates for fixed field sizes.

Contribution

It extends the capacity formulas of the AMMC to cases where $2n ot extless m$, providing more general results and improved error bounds for fixed finite fields.

Findings

Capacity formula now applies without the $2n extless m$ restriction.

Improved error term for fixed finite field size.

Generalization of asymptotic capacity results to broader parameter ranges.

Abstract

The Additive-Multiplicative Matrix Channel (AMMC) was introduced by Silva, Kschischang and K\"otter in 2010 to model data transmission using random linear network coding. The input and output of the channel are $n\times m$ matrices over a finite field $\mathbb{F}_q$. On input the matrix $X$, the channel outputs $Y=A(X+W)$ where $A$ is a uniformly chosen $n\times n$ invertible matrix over $\mathbb{F}_q$ and where $W$ is a uniformly chosen $n\times m$ matrix over $\mathbb{F}_q$ of rank $t$. Silva \emph{et al} considered the case when $2n\leq m$. They determined the asymptotic capacity of the AMMC when $t$, $n$ and $m$ are fixed and $q\rightarrow\infty$. They also determined the leading term of the capacity when $q$ is fixed, and $t$, $n$ and $m$ grow linearly. We generalise these results, showing that the condition $2n\geq m$ can be removed. (Our formula for the capacity falls into two cases, one of which generalises the $2n\geq m$ case.) We also improve the error term in the case when $q$ is fixed.