Distance bounds for convolutional codes and some optimal codes

TL;DR

This paper discusses bounds on the distance of convolutional codes, presents codes meeting these bounds, and explores the field size requirements for MDS convolutional codes, including examples of cyclic codes.

Contribution

It introduces new codes meeting distance bounds, derives a lower bound for field size of MDS convolutional codes, and provides examples of cyclic convolutional codes.

Findings

Several codes meet the Griesmer and Heller bounds.

The lower bound for field size of MDS convolutional codes is often tight.

Most examples are cyclic convolutional codes, illustrating their properties.

Abstract

After a discussion of the Griesmer and Heller bound for the distance of a convolutional code we present several codes with various parameters, over various fields, and meeting the given distance bounds. Moreover, the Griesmer bound is used for deriving a lower bound for the field size of an MDS convolutional code and examples are presented showing that, in most cases, the lower bound is tight. Most of the examples in this paper are cyclic convolutional codes in a generalized sense as it has been introduced in the seventies. A brief introduction to this promising type of cyclicity is given at the end of the paper in order to make the examples more transparent.