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In this paper we introduce a binomial ideal derived from a binary linear code. We present some applications of a Gröbner basis of this ideal with respect to a total degree ordering. In the first application we give a decoding method for the code. By associating the code with the set of cycles in a graph, we can solve the problem of finding all codewords of… (More)

This paper addresses the question of how often the square code of an arbitrary l-dimensional subcode of the code GRS k (a, b) is exactly the code GRS 2k−1 (a, b * b). To answer this question we first introduce the notion of gaps of a code which allows us to characterize such subcodes easily. This property was first stated and used in [10] where Wieschebrink… (More)

In this paper we introduce the class of composite access structures for secret sharing. We also provide secret sharing schemes realizing these structures and study their information rates. As a particular case of this construction, we present the subclass of iterated threshold schemes, a large class of ideal secret sharing schemes.

This paper addresses the question of retrieving the triple

We show herein that a pattern based on FGLM techniques can be used for computing Gröbner bases, or related structures, associated to linear codes. This Gröbner bases setting turns out to be strongly related to the combinatorics of the codes.

Code-based cryptography is an interesting alternative to classic number-theoretic public key cryptosys-tem since it is conjectured to be secure against quantum computer attacks. Many families of codes have been proposed for these cryptosystems such as algebraic geometry codes. In (Márquez-Corbella et al., 2012) — for so called very strong algebraic geometry… (More)

—There are two gradient descent decoding procedures for binary codes proposed independently by Liebler and by Ashikhmin and Barg. Liebler in his paper [15] mentions that both algorithms have the same philosophy but in fact they are rather different. The purpose of this communication is to show that both algorithms can be seen as two ways of understanding… (More)