Decoding convolutional codes over finite rings. A linear dynamical systems approach

TL;DR

This paper presents a linear dynamical systems approach to decoding convolutional codes over finite rings, enabling efficient decoding algorithms and analysis of their error-correction capabilities.

Contribution

It introduces a novel approach using linear dynamical systems for decoding convolutional codes over Zpr and combines existing algorithms for improved decoding performance.

Findings

Decoding over Zpr reduces to algorithms over finite fields.

The combined decoding algorithm has quantifiable error-correction capabilities.

The approach offers insights into the time complexity of decoding.

Abstract

Observable convolutional codes defined over Zpr with the Predictable Degree Property admit minimal input/state/output representations that preserve structural properties under scalar restriction. We make use of this fact to present Rosenthal's decoding algorithm for these convolutional codes. When combined with the Greferath-Vellbinger algorithm and a modified version of the Torrecillas-Lobillo-Navarro algorithm, the decoding problem of convolutional codes over Zpr reduces to selecting two decoding algorithms for linear block codes over a field. Finally, we analyze both the theoretical and practical error-correction capabilities of the combined algorithm as well as its time complexity.