Construction and Decoding of Convolutional Codes with optimal Column Distances

TL;DR

This paper introduces a new construction method for convolutional codes with optimal column distances over arbitrary finite fields, proving their uniqueness and relating their structure to Reed-Muller codes, while also developing a reduced complexity decoding algorithm.

Contribution

The paper presents a novel construction of convolutional codes with optimal column distances and proves their uniqueness, linking their structure to Reed-Muller codes and improving decoding efficiency.

Findings

Constructed convolutional codes with optimal column distances over arbitrary finite fields.

Proved the uniqueness of these codes for given parameters.

Developed a reduced complexity Viterbi decoding algorithm for the codes.

Abstract

The construction of Maximum Distance Profile (MDP) convolutional codes in general requires the use of very large finite fields. In contrast convolutional codes with optimal column distances maximize the column distances for a given arbitrary finite field. In this paper, we present a construction of such convolutional codes. In addition, we prove that for the considered parameters the codes that we constructed are the only ones achieving optimal column distances. The structure of the presented convolutional codes with optimal column distances is strongly related to first order Reed-Muller block codes and we leverage this fact to develop a reduced complexity version of the Viterbi algorithm for these codes.