Abstract – A general algebraic method for decoding all cyclic codes up to their actual minimum distance d is presented. Full error-correcting capabilities t = [(d − 1)/2] of the codes are therefore achieved. In contrast to the decoding method recently suggested by Chen et. al., our method uses for the first time characteristic sets instead of Gröbner bases as the algebraic tool to solve the system of multivariate syndrome equations. The characteristic sets method is generally faster than the Gröbner bases method.

A new strategy called “Fill-Holes” method is also presented. It uses Gröbner bases or characteristic sets to find certain unknown syndromes and then combines the computational methods with the well-implemented BCH decoding algorithm.

Keywords – Coding theory, cyclic codes, decoding, ideal theory, Gröbner bases, characteristic sets.

INTRODUCTION

One important objective in coding theory has always been the construction of algebraic algorithms, that are capable of decoding all cyclic codes up to their actual minimum distance. Full error-correcting capabilities of the codes can only be achieved when such algorithms are available. For many years, algebraic decoding of cyclic codes has been constrained by the lower bound on the minimum distance of the codes. For example, the commonly used Berlekamp-Massey algorithm is known to be restricted within the BCH bound when it is used to decode all cyclic codes. Such restrictions can be traced to the fact that the algorithm requires syndromes to be contiguous in the Newton's identities.