Generalized column distances

TL;DR

This paper introduces the concept of r-generalized column distances for convolutional codes, extending existing invariants and analyzing their properties as the code length increases.

Contribution

It defines and studies the properties of r-generalized column distances for convolutional codes, providing a new perspective on their invariants.

Findings

Defined r-generalized column distances for truncated codes

Established properties of these invariants

Compared with existing invariants in the literature

Abstract

We define a notion of r-generalized column distances for the j-truncation of a convolutional code. Taking the limit as j tends to infinity allows us to define r-generalized column distances of a convolutional code. We establish some properties of these invariants and compare them with other invariants of convolutional codes which appear in the literature.