In this note we give a theoretical support by means of quotient polynomial rings for the computation formulas of the dimension of abelian codes.

In this paper, we extend results about the implementation of the Berlekamp-Massey-Sakata algorithm on data tables having a number of unknown values.

We study the problem of the computation of Groebner basis for the ideal of linear recurring relations of a doubly periodic array. We find a set of indexes such that, along with some conditions, guarantees that the set of polynomials obtained at the last iteration in the Berlekamp-Massey-Sakata algorithm is exactly a Groebner basis for the mentioned ideal. Then, we apply these results to improve locator decoding in abelian codes.

In this paper, we develop a technique to extend any bound for the minimum distance of cyclic codes constructed from its defining sets (ds-bounds) to Abelian (or multivariate) codes through the notion of B-apparent distance. We also study conditions for an Abelian code to verify that its B-apparent distance reaches its (true) minimum distance. Then, we construct some codes as an application.

Reed-Muller codes belong to the family of affine-invariant codes. As such codes, they have a defining set that determines them uniquely, and they are extensions of cyclic group codes. In this paper, we identify those cyclic codes with multidimensional abelian codes and we use the techniques introduced by Bernal and Simón to construct information sets for them from their defining set. For first- and second-order Reed-Muller codes, we describe a direct method to construct information sets in terms of their basic parameters.

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.

In our earlier work, we introduced a technique to construct an information set for every semisimple Abelian code over an arbitrary field, solely in terms of its defining set. In this paper, we apply the geometrical properties of those information sets to obtain sufficient conditions for a -error correcting Abelian code to have a b-PD-set for every b ≤ t. These conditions are simply given in terms of the structure of the defining set of the code.

We show some sufficient conditions for a semisimple abelian code to be permutation decodable. We find non trivial partial PD-sets with respect to the information set introduced red by the authors in a previous paper. By using these results we present examples of permutation decodable abelian codes improving the bounds about the dimension given by S. G. S. Shiva, K. C. Fung and H. S. Y. Tan.

We describe a technique to construct a set of check positions (and hence an information set) for every abelian code solely in terms of its defining set. This generalizes that given by Imai in in the case of binary TDC codes.

The paper present a method based on the computation of the cardinalities of certain cyclotomic cosets on different extensions of the ground field. Such cosets are completely determined by the structure of the defining set of the code. This method agrees with that given in the works of Imai in the case of two dimensional binary (TDC) codes. This technique allows us to take some advantages. On the one hand, the codes were constructed with prescribed information sets in order to get particular properties that we may need; on the other hand, in the case of a fixed code, it allows to determine, a priori, the shape of our information set. It may be crucial, for example, in order to make a permutation decoding.