论文标题
有限场矩阵通道的容量
The capacity of a finite field matrix channel
论文作者
论文摘要
Silva，Kschischang和Kötter于2010年引入了添加剂 - 型矩阵通道（AMMC），以使用随机线性网络编码来建模数据传输。通道的输入和输出为有限字段$ \ mathbb {f} _q $上的$ n \ times m $矩阵。在输入矩阵$ x $上，通道输出$ y = a（x+w）$，其中$ a $是均匀选择的$ n \ times n $ n $ n $可转化矩阵，超过$ \ mathbb {f} _q $，其中$ w $是均匀选择的$ n \ times $ n \ times m $ matrix，而不是$ \ mathbb $ \ mathbb $ \ f} $。 Silva \ emph {et al}考虑了$ 2N \ leq m $的情况。当$ t $，$ n $和$ m $固定以及$ q \ rightarrow \ infty $时，他们确定了AMMC的渐近容量。他们还确定了固定$ q $的前期术语，以及$ t $，$ n $和$ m $线性的线性生长。我们概括了这些结果，表明可以删除条件$ 2N \ geq m $。 （我们的容量公式分为两种情况，其中一种概括了$ 2N \ geq m $案例。）在固定$ q $的情况下，我们还提高了错误期限。
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.
