Generalized weights of convolutional codes

TL;DR

This paper introduces a new definition for generalized weights of convolutional codes that considers their module structure, providing bounds, classifications, and relations to previous definitions.

Contribution

It proposes a novel module-based definition of generalized weights for convolutional codes, extending prior work and analyzing their properties and bounds.

Findings

Established upper bounds for MDS and MDP codes

Classified optimal anticodes and computed their weight hierarchy

Linked new weights with code parameters like length, rank, and degree

Abstract

In 1997 Rosenthal and York defined generalized Hamming weights for convolutional codes, by regarding a convolutional code as an infinite dimensional linear code endowed with the Hamming metric. In this paper, we propose a new definition of generalized weights of convolutional codes, that takes into account the underlying module structure of the code. We derive the basic properties of our generalized weights and discuss the relation with the previous definition. We establish upper bounds on the weight hierarchy of MDS and MDP codes and show that that, depending on the code parameters, some or all of the generalized weights of MDS codes are determined by the length, rank, and internal degree of the code. We also prove an anticode bound for convolutional codes and define optimal anticodes as the codes which meet the anticode bound. Finally, we classify optimal anticodes and compute their weight hierarchy.