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.

翻译：Additive-Multiplicative Matrix Channel (AMMC) 由 Silva, Kschischang 和 Kötter 于 2010 年提出，用于建模基于随机线性网络编码的数据传输。该信道的输入和输出为有限域 \(\mathbb{F}_q\) 上的 \(n \times m\) 矩阵。输入矩阵 \(X\) 时，信道输出 \(Y = A(X + W)\)，其中 \(A\) 为均匀选取的 \(\mathbb{F}_q\) 上的 \(n \times n\) 可逆矩阵，\(W\) 为均匀选取的秩为 \(t\) 的 \(\mathbb{F}_q\) 上的 \(n \times m\) 矩阵。Silva 等人考虑了 \(2n \leq m\) 的情形，确定了当 \(t, n, m\) 固定且 \(q \to \infty\) 时 AMMC 的渐近容量，并给出了当 \(q\) 固定且 \(t, n, m\) 线性增长时容量的主导项。本文推广了这些结果，证明可去除 \(2n \geq m\) 的条件（容量公式分为两种情况，其中一种推广了 \(2n \geq m\) 情形），同时改进了 \(q\) 固定时的误差项。