Gröbner Bases and Coding Theory

Summary

Abstract

In this note, we present a sketch of several interplays between Gröbner bases theory and coding theory. For readers who are not so familiar to coding theory, some introductory explanations on error-correcting codes and coding theory are included. The main topics are some problems of encoding and decoding of algebraic codes which are related to Gröbner bases. Some simple examples of codes are referred to. In particular, recent developments in coding theory which have been done around multidimensional or multivariate codes have initiated and strengthened the connections, and several new problems and relevant algorithms have been explored in coding theory.

Introduction

In this note, we give a sketch of several interactions between coding theory (or rather algebraic coding theory) and Gröbner bases theory. Precisely, coding theory is the theory of error-correcting codes, which are used widely in digital communication and storage systems to transmit or to store digital information error-free or as correct as possible. Although error-correcting codes are probabilistic by nature in the sense that they are used to battle against random noise in transmission or storage channels, practical codes are constructed in deterministic fashions with algebraic methods, without which error control cannot be achieved with low complexity and cost. As a result, almost all of them are algebraic. Therefore we discuss nothing but the algebraic error-correcting codes.