Related papers: A decoding algorithm for binary linear codes using…
There are several efficient methods to solve linear interval polynomial systems in the context of interval computations, however, the general case of interval polynomial systems is not yet covered as well. In this paper we introduce a new…
We formulate the classical decoding algorithm of alternant codes afresh based on interpolation as in Sudan's list decoding of Reed-Solomon codes, and thus get rid of the key equation and the linear recurring sequences in the theory. The…
In this contribution, we consider a zero-dimensional polynomial system in $n$ variables defined over a field $\mathbb{K}$. In the context of computing a Rational Univariate Representation (RUR) of its solutions, we address the problem of…
Decoding a Reed-Solomon code can be modeled by a bilinear system which can be solved by Gr\"obner basis techniques. We will show that in this particular case, these techniques are much more efficient than for generic bilinear systems with…
Proving statements about linear operators expressed in terms of identities often leads to finding elements of certain form in noncommutative polynomial ideals. We illustrate this by examples coming from actual operator statements and…
Recent advancements in quantum computing and quantum-inspired algorithms have sparked renewed interest in binary optimization. These hardware and software innovations promise to revolutionize solution times for complex problems. In this…
We adress the problem of the algebraic decoding of any cyclic code up to the true minimum distance. For this, we use the classical formulation of the problem, which is to find the error locator polynomial in terms of the syndroms of the…
A contemporary and exciting application of Groebner bases is their use in computational biology, particularly in the reverse engineering of gene regulatory networks from experimental data. In this setting, the data are typically limited to…
In this paper we describe an efficient involutive algorithm for constructing Groebner bases of polynomial ideals. The algorithm is based on the concept of involutive monomial division which restricts the conventional division in a certain…
In this article we present two new algorithms to compute the Groebner basis of an ideal that is invariant under certain permutations of the ring variables and which are both implemented in SINGULAR (cf. [DGPS12]). The first and major…
Practically good error-correcting codes should have good parameters and efficient decoding algorithms. Some algebraically defined good codes such as cyclic codes, Reed-Solomon codes, and Reed-Muller codes have nice decoding algorithms.…
We investigate the structure of ideals generated by binomials (polynomials with at most two terms) and the schemes and varieties associated to them. The class of binomial ideals contains many classical examples from algebraic geometry, and…
Binomial ideals are special polynomial ideals with many algorithmically and theoretically nice properties. We discuss the problem of deciding if a given polynomial ideal is binomial. While the methods are general, our main motivation and…
In this paper, we mainly study quaternary linear codes and their binary subfield codes. First we obtain a general explicit relationship between quaternary linear codes and their binary subfield codes in terms of generator matrices and…
We consider the problem of inference in higher-order undirected graphical models with binary labels. We formulate this problem as a binary polynomial optimization problem and propose several linear programming relaxations for it. We compare…
This paper is concerned with linear algebra based methods for solving exactly polynomial systems through so-called Gr\"obner bases, which allow one to compute modulo the polynomial ideal generated by the input equations. This is a topical…
This work introduces a decoding strategy for binary self-dual codes possessing an automorphism of a specific type. The proposed algorithm is a hard decision iterative decoding scheme. The enclosed experiments show that the new decoding…
We prove that the log canonical thresholds of a large class of binomial ideals, such as complete intersection binomial ideals and the defining ideals of space monomial curves, are computable by linear programming.
A computation method of algebraic local cohomology with parameters, associated with zero-dimensional ideal with parameter, is introduced. This computation method gives us in particular a decomposition of the parameter space depending on the…
Due to the elimination property held by the lexicographic monomial order, the corresponding Groebner bases display strong structural properties from which meaningful informations can easily be extracted. We study these properties for…