Algebra in Algorithmic Coding Theory

TL;DR

This paper surveys the role of algebra in the development of error-correcting codes and algorithms, highlighting both historical context and modern algebraic constructions to improve information transmission reliability.

Contribution

It clarifies the often underappreciated role of algebra in designing algorithms for error-correcting codes, bridging the gap between algebraic theory and practical coding algorithms.

Findings

Overview of error-correcting codes and their algebraic foundations

Description of modern algebraic constructions for codes

Explanation of algebraic algorithms for code implementation

Abstract

We survey the notion and history of error-correcting codes and the algorithms needed to make them effective in information transmission. We then give some basic as well as more modern constructions of, and algorithms for, error-correcting codes that depend on relatively simple elements of applied algebra. While the role of algebra in the constructions of codes has been widely acknowledged in texts and other writings, the role in the design of algorithms is often less widely understood, and this survey hopes to reduce this difference to some extent.