  We deal with two problems related with the use of the Sakata's algorithm in a
specific class of bivariate codes. The first one is to improve the general
framework of locator decoding in order to apply it on such abelian codes. The
second one is to find a set of indexes oF the syndrome table such that no other
syndrome contributes to implement the BMSa and, moreover, any of them may be
ignored \textit{a priori}. In addition, the implementation on those indexes is
sufficient to get the Groebner basis; that is, it is also a termination
criterion.
