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Computer Science > Information Theory
Title: The capacity of a finite field matrix channel
(Submitted on 25 Oct 2022 (v1), last revised 26 Apr 2024 (this version, v3))
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.
Submission history
From: Simon Blackburn [view email][v1] Tue, 25 Oct 2022 15:36:28 GMT (18kb)
[v2] Tue, 29 Nov 2022 12:58:36 GMT (18kb)
[v3] Fri, 26 Apr 2024 14:11:44 GMT (20kb)
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