Distance bounds for convolutional codes and some optimal codes

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.