Optimal Multidimensional Convolutional Codes
2026-03-25 • Information Theory
Information Theory
AI summaryⓘ
The authors study multi-dimensional (mD) convolutional codes, which are an extension of common one-dimensional convolutional codes to higher dimensions. They focus on codes with finite support that work over finite fields and have a rate and degree defining their structure. The authors recall an important upper limit called the mD generalized Singleton bound that restricts the error-correcting ability of these codes. Their main contribution is creating new examples of optimal mD convolutional codes, called MDS codes, which meet this bound by using special matrices with a property called superregularity. This work broadens the known methods for building such highly efficient multi-dimensional codes.
m-dimensional convolutional codesfinite supportfinite fieldsfree distancegeneralized Singleton boundmaximum distance separable (MDS) codessuperregular matricesgenerator matricescode ratecode degree
Authors
Z. Abreu, J. Lieb, R. Pinto, R. Simoes
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~$δ$, 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.