  General error locator polynomials are polynomials able to decode any
correctable syndrome for a given linear code. Such polynomials are known to
exist for all cyclic codes and for a large class of linear codes. We provide
some decoding techniques for affine-variety codes using some multidimensional
extensions of general error locator polynomials. We prove the existence of such
polynomials for any correctable affine-variety code and hence for any linear
code. We propose two main different approaches, that depend on the underlying
geometry. We compute some interesting cases, including Hermitian codes. To
prove our coding theory results, we develop a theory for special classes of
zero-dimensional ideals, that can be considered generalizations of stratified
ideals. Our improvement with respect to stratified ideals is twofold: we
generalize from one variable to many variables and we introduce points with
multiplicities.
