The capacity of a finite field matrix channel

TL;DR

This work determines the capacity of the Additive-Multiplicative Matrix Channel (AMMC) over finite fields, removing the previous constraint $2n\le m$ and identifying the capacity in two regimes: if $m+t\ge 2n$ the capacity tends to $$(m-n)(n-t)$, and otherwise to $\frac{(m-t)^2}{4}$$, with the results holding in both $q\to\infty$ (fixed $(n,m,t,k)$) and fixed-$q$ (linear growth) settings. The analysis reduces the problem to the $k$-AMMC (input rank $k$) and shows the capacity-achieving input is uniform over rank-$k$ matrices; the AMMC capacity is the maximum over $k$ of the $k$-AMMC capacities. A practical coding scheme achieving capacity in the asymptotic regimes is provided, based on a block-structured encoding that leverages row-space properties and row-reduction at the receiver, and the work connects the AMMC to subspace/operator-channel models to enable subspace decoding techniques. The results corroborate the Blackburn–Claridge conjecture, sharpen error terms in the fixed-field regime, and enhance the understanding of RLNC-motivated matrix channels with both errors and random network coding transfers.

Abstract

The Additive-Multiplicative Matrix Channel (AMMC) was introduced by Silva, Kschischang and Kötter 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.

Figures (1)

• Figure 1: An example of random network coding. Here we are using binary vectors of length $6$. Source node $X$ sends (say) $x_1=(100000), x_2=(010000), x_3=(001000), x_4=(101000), x_5=(110000)$, and $x_6=(111000)$ to nodes $U_i$ as shown. Nodes $U_1$ and $U_2$ send linear combinations of received vectors to $V$, where $u_1=x_1+x_2$, $u_2=x_2$, $u_3=x_4$ and $u_4=x_3+x_4$. The node $U_3$ is faulty, and sends random vectors $u_5=x_5+(101000)$ and $u_6=x_6+(110011)$ to $V$. Finally, node $V$ sends the vectors $y_i$ to the sink node $Y$, where $y_1=u_1+u_3+u_5$, $y_2=u_2+u_5+u_6$, $y_3=u_1+u_3+u_4+u_5$, $y_4=u_5+u_6$, $y_5=u_3+u_4+u_6$ and $y_6=u_2+u_6$.

Theorems & Definitions (27)

• Lemma 1
• Corollary 2
• proof
• Corollary 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6