Optimal Multidimensional Convolutional Codes

TL;DR

This paper extends the theory of convolutional codes to multiple dimensions, establishing bounds and constructing new classes of optimal codes that achieve maximum distance separation in higher dimensions.

Contribution

It introduces new constructions of MDS $m$D convolutional codes based on superregular matrices, expanding the known set of such optimal codes.

Findings

Developed the $m$D generalized Singleton bound for convolutional codes.

Constructed new families of MDS $m$D convolutional codes with rate $1/n$.

Expanded the theoretical framework and practical constructions of optimal multidimensional codes.

Abstract

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~$\delta$, 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.