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.