In this paper, we analyze $m$-dimensional ($m$D) convolutional codes with finite support, viewed as a natural generalization of one-dimensional (1D) convolutional codes to higher dimensions. An $m$D convolutional code with finite support consists of codewords with compact support indexed in $\mathbb{N}^m$ and taking values in $\mathbb{F}_{q}[z_1,\ldots,z_m]^n$, where $\mathbb{F}_{q}$ is a finite field with $q$ elements. We recall a natural upper bound on the free distance of an $m$D convolutional code with rate $k/n$ and degree~$δ$, called $m$D generalized Singleton bound. Codes that attain this bound are called maximum distance separable (MDS) $m$D convolutional codes. As our main result, we develop new constructions of MDS $m$D convolutional codes based on superregularity of certain matrices. Our results include the construction of new families of MDS $mD$ convolutional codes of rate $1/n$, relying on generator matrices with specific row degree conditions. These constructions significantly expand the set of known constructions of MDS $m$D convolutional codes.

翻译：本文分析具有有限支撑的$m$维（$m$D）卷积码，将其视为一维（1D）卷积码向高维的自然推广。具有有限支撑的$m$D卷积码由索引在$\mathbb{N}^m$中且取值于$\mathbb{F}_{q}[z_1,\ldots,z_m]^n$的紧支撑码字构成，其中$\mathbb{F}_{q}$是含有$q$个元素的有限域。我们回顾了速率为$k/n$、度为$\delta$的$m$D卷积码自由距离的自然上界，即$m$D广义Singleton界。达到该界的码称为最大距离可分（MDS）$m$D卷积码。作为主要结果，我们基于特定矩阵的超规则性，提出了MDS $m$D卷积码的新构造方法。我们的结果包括构造速率为$1/n$的MDS $m$D卷积码新族，其依赖于具有特定行度条件的生成矩阵。这些构造显著扩展了已知MDS $m$D卷积码的构造集合。