Information sets in abelian codes: defining sets and Groebner basis

In [BS2] we introduced a technique to construct information sets for every semisimple abelian code by means of its defining set. This construction is a non trivial generalization of that given by H. Imai [Im] in the case of binary two-dimensional cyclic (TDC) codes. On the other hand, S. Sakata [Sak] showed a method for constructing information sets for binary TDC codes based on the computation of Groebner basis which agrees with the information set obtained by Imai. Later, H.Chabanne [Chab] presents a generalization of the permutation decoding algorithm for binary abelian codes by using Groebner basis, and as a part of his method he constructs an information set following the same ideas introduced by Sakata. In this paper we show that, in the general case of q-ary multidimensional abelian codes, both methods, that based on Groebner basis and that defined in terms of the defining sets, also yield the same information set.