This paper is devoted to studying two main problems: 1) computing the apparent distance of an Abelian code and 2) giving a notion of Bose, Ray-Chaudhuri, Hocquenghem (BCH) multivariate code. To do this, we first strengthen the notion of an apparent distance by introducing the notion of a strong apparent distance; then, we present an algorithm to compute the strong apparent distance of an Abelian code, based on some manipulations of hypermatrices associated with its generating idempotent. Our method uses less computations than those given by Camion and Sabin; furthermore, in the bivariate case, the order of computation complexity is reduced from exponential to linear. Then, we use our techniques to develop a notion of a BCH code in the multivariate case, and we extend most of the classical results on cyclic BCH codes. Finally, we apply our method to the design of Abelian codes with maximum dimension with respect to a fixed apparent distance and a fixed length.

The P. Camion's apparent distance of an abelian code is a generalization of the notion of the BCH bound of cyclic codes [3]. In this work, we present a method of computation of the apparent distance in multivariate abelian codes, based on manipulations of hypermatrices. Our algorithm needs fewer computations than any other, up to our knowledge; in fact, in the case of two dimensional abelian codes it has linear complexity. We give two applications. First, we construct abelian codes that multiply the dimension of a given cyclic code and equal its BCH bound. The second one is an approximation to a notion of BCH multivariate code.