Construction of Unit-Memory MDS Convolutional Codes

TL;DR

This paper presents a new algebraic construction of unit-memory MDS convolutional codes over small fields, expanding the family of optimal codes with flexible parameters and introducing many new strongly-MDS codes.

Contribution

The paper provides a large family of unit-memory MDS convolutional codes over smaller fields, with novel algebraic construction methods and new subclasses of strongly-MDS codes.

Findings

Constructed a large family of unit-memory MDS convolutional codes

Reduced the field size needed for code construction

Generated many new strongly-MDS convolutional codes

Abstract

Maximum-distance separable (MDS) convolutional codes form an optimal family of convolutional codes, the study of which is of great importance. There are very few general algebraic constructions of MDS convolutional codes. In this paper, we construct a large family of unit-memory MDS convolutional codes over $\F$ with flexible parameters. Compared with previous works, the field size $q$ required to define these codes is much smaller. The construction also leads to many new strongly-MDS convolutional codes, an important subclass of MDS convolutional codes proposed and studied in \cite{GL2}. Many examples are presented at the end of the paper.