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 delta, that have minimum distance d(C)= delta. 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.