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Given a local ring (R , m) and an elliptic curve E (R / m) , we define its elliptic loop as the points of P 2 (R) projecting to E under the canonical modulo- m reduction, endowed with an operation that extends the curve's addition. While its subset of points satisfying the curve's Weierstrass equation is a group, this larger object is proved to be a power-associative abelian algebraic loop, which is seldom completely associative. When an elliptic loop has no points of order 3, its affine part is obtained as a stratification of a one-parameter family of elliptic curves defined over R , which we call layers. Stronger associativity properties are established when m e vanishes for small values of e ∈ Z. When the underlying ring is R = Z / p e Z , the infinity part of an elliptic loop is generated by two elements, the group structure of layers may be established and the points with the same projection and same order possess a geometric description. [ABSTRACT FROM AUTHOR]

Abstract: We propose a new syndrome variety, which can be used to decode cyclic codes. We present also a generalization to erasure and error decoding. We can exhibit a polynomial whose roots give the error locations, once it has been specialized to a given syndrome. This polynomial has degree in the variable corresponding to the error locations and its coefficients are polynomials in the syndromes. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]