Cyclic and BCH Codes whose Minimum Distance Equals their Maximum BCH bound

Abstract

In this paper we study the family of cyclic codes such that its minimum distance reaches the maximum of its BCH bounds. We also show a way to construct cyclic codes with that property by means of computations of some divisors of a polynomial of the form X^n −1. We apply our results to the study of those BCH codes C, with designed distance δ, that have minimum distance d(C) = δ. Finally, we present some examples of new binary BCH codes satisfying that condition. To do this, we make use of two related tools: the discrete Fourier transform and the notion of apparent distance of a code, originally defined for multivariate abelian codes.