Apparent distance and a notion of BCH multivariate codes

Abstract

This paper is devoted to study two main problems: on the one hand, to compute the apparent distance of an abelian code and on the other hand, to give a notion of BCH multivariate code. To do this, we present an algorithm to compute the apparent distance of an abelian code, based on some manipulations of hypermatrices associated to its generating idempotent. Our method uses less computations than those given in [5] and [10]; furthermore, in the bivariate case, the order of the computations is reduced from exponential to linear. Then we use our techniques to develop a notion of BCH code in the multivariate case and we extend most of the classical results on BCH codes. Finally, we apply our method in two directions: we construct abelian codes from cyclic codes, multiplying their dimension and preserving their apparent distance; and we design abelian codes with maximum dimension with respect to a fixed apparent distance and a fixed length.