We present a structure associated to the class of linear codes. The properties of that structure are similar to some structures in the linear algebra techniques into the framework of the Gröbner bases tools. It allows to get some insight in the problem of determining whether two codes are permutation equivalent or not. Also an application to the decoding problem is presented, with particular emphasis on the binary case.
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.
Two correspondences have been provided that associate any linear code over a finite field with a binomial ideal. In this paper, algorithms for computing their Graver bases and universal Gröbner bases are given. To this end, a connection between these binomial ideals and toric ideals will be established.
Each linear code can be described by a code ideal given as the sum of a toric ideal and a non-prime ideal. In this way, several concepts from the theory of toric ideals can be translated into the setting of code ideals. It will be shown that after adjusting some of these concepts, the same inclusion relationship between the set of circuits, the universal Gröbner basis and the Graver basis holds. Furthermore, in the case of binary linear codes, the universal Gröbner basis will consist of all binomials which correspond to codewords that satisfy the Singleton bound and a particular rank condition…
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 minimal length (minimal cycles in a graph), and show how to find a minimal cycle basis. Finally we discuss some results on the computation of the Gröbner basis.
The brain processes information about the environment via neural codes. The neural ideal was introduced recently as an algebraic object that can be used to better understand the combinatorial structure of neural codes. Every neural ideal has a particular generating set, called the canonical form, that directly encodes a minimal description of the receptive field structure intrinsic to the neural code. On the other hand, for a given monomial order, any polynomial ideal is also generated by its unique (reduced) Gröbner basis with respect to that monomial order. How are these two types of generat…
Generalized quasi-cyclic (GQC) codes form a wide and useful class of linear codes that includes thoroughly quasi-cyclic codes, finite geometry (FG) low density parity check (LDPC) codes, and Hermitian codes. Although it is known that the systematic encoding of GQC codes is equivalent to the division algorithm in the theory of Grobner basis of modules, there has been no algorithm that computes Grobner basis for all types of GQC codes. In this paper, we propose two algorithms to compute Grobner basis for GQC codes from their parity check matrices: echelon canonical form algorithm and transpose a…
In 1980, White conjectured that the toric ideal of a matroid is generated by quadratic binomials corresponding to a symmetric exchange. In this paper, we compute Gröbner bases of toric ideals associated with matroids and show that, for every matroid on ground sets of size at most seven except for two matroids, Gröbner bases of toric ideals consist of quadratic binomials corresponding to a symmetric exchange.
In the present paper, we prove that the toric ideals of certain $s$-block diagonal matching fields have quadratic Gröbner bases. Thus, in particular, those are quadratically generated. By using this result, we provide a new family of toric degenerations of Grassmannians.
This paper deals with the notion of Gröbner $δ$-base for some rings of linear differential operators by adapting the works of W. Trinks, A. Assi, M. Insa and F. Pauer. We compare this notion with the one of Gröbner base for such rings. As an application, and following a previous work of A. Assi, we give some results on finiteness and on flatness of finitely generated left modules over these rings.
We characterise the class of one-cogenerated Pfaffian ideals whose natural generators form a Gröbner basis with respect to any anti-diagonal term-order. We describe their initial ideals as well as the associated simplicial complexes, which turn out to be shellable and thus Cohen-Macaulay. We also provide a formula for computing their multiplicity.
In the article "Non-commutative Grobner bases for commutative algebras", Eisenbud-Peeva-Sturmfels proved a number of results regarding Grobner bases and initial ideals of those ideals in the free associative algebra which contain the commutator ideal. We prove similar results for ideals which contains the anti-commutator ideal (the defining ideal of the exterior algebra). We define one notion of generic initial ideals in the free assoicative algebra, and show that gin's of ideals containing the commutator ideal, or the anti-commutator ideal, are finitely generated.
This paper is a survey on the study of the behaviour of the composition of polynomials on the computation of Gröbner bases. This survey brings together some works published between 1995 and 2007. The authors of these papers gave answers to some questions in this subject for several types of Gröbner bases, over different monomials orderings and over different polynomials rings. Some of these answers are complete and some are not. Some papers of them were given to answer some open questions given in the first published paper of these papers and other some were given to generalize previous work. …
We present basic properties of Gröbner bases of submodules of a free module of finite rank over a polynomial ring $R$ with coefficients in a graded truncated discrete valuations ring $A$. As an application, we give a criterion for a finitely generated $R$-module to be flat over $A$. Its non-graded version is also given.
We present an $m$-adic Newton iteration with quadratic convergence for lexicographic Gröbner basis of zero dimensional ideals in two variables. We rely on a structural result about the syzygies in such a basis due to Conca and Valla, that allowed them to explicitly describe these Gröbner bases by affine parameters; our Newton iteration works directly with these parameters.
Chiba University（千叶大学）；Zuse Institute Berlin（柏林Zuse研究所）；Nihon University（日本大学）；Mitsubishi Electric Information Technology R&D Center（三菱电机信息技术研发中心）；Université de Limoges（利摩日大学）；Rikkyo University (St. Paul’s University)（立教大学（圣保罗大学））
D. Bayer and M. Stillman showed that Grobner bases can be used to compute the Castelnuovo-Mumford regularity, which is a measure for the vanishing of graded local cohomology modules. The aim of this paper is to show that the same method can be applied to study other cohomological invariants as well as the reduction number.
Taylor presented an explicit resolution for arbitrary monomial ideals. Later, Lyubeznik found that already a subcomplex defines a resolution. We show that the Taylor resolution may be obtained by repeated application of the Schreyer Theorem from the theory of Grobner bases, whereas the Lyubeznik resolution is a consequence of Buchberger's chain criterion. Finally, we relate Froberg's contracting homotopy for the Taylor complex to normal forms with respect to our Grobner bases and use it to derive a splitting homotopy that leads to the Lyubeznik complex.
We establish the first previously unknown case of the Eisenbud-Harris conjecture in Castelnuovo theory concerning algebraic curves of high genus in ${\bf P}^n$. The problem is reduced to a question about zero-dimensional schemes $Γ\subset {\bf P}^{n-1}$ in symmetric position with certain constrains on the Hilbert function. The method of Gröbner bases is then applied to study the homogeneous ideal of $Γ$.
In this paper we develop a Grobner bases theory for ideals of partial difference polynomials with constant or non-constant coefficients. In particular, we introduce a criterion providing the finiteness of such bases when a difference ideal contains elements with suitable linear leading monomials. This can be explained in terms of Noetherianity of the corresponding quotient algebra. Among these Noetherian quotients we find finitely generated polynomial algebras where the action of suitable finite dimensional commutative algebras and in particular finite abelian groups is defined. We obtain ther…
We improve certain degree bounds for Grobner bases of polynomial ideals in generic position. We work exclusively in deterministically verifiable and achievable generic positions of a combinatorial nature, namely either strongly stable position or quasi stable position. Furthermore, we exhibit new dimension- (and depth-)dependent upper bounds for the Castelnuovo-Mumford regularity and the degrees of the elements of the reduced Grobner basis (w.r.t. the degree reverse lexicographical ordering) of a homogeneous ideal in these positions.
In this article we consider the problem of finding generators of the toric ideal associated to a combinatorial object called a staged tree. Our main theorem states that toric ideals of staged trees that are balanced and stratified are generated by a quadratic Gröbner basis whose intial terms are squarefree. The proof of this result is based on Sullivant's toric fiber product construction.
We present results, both old and new, concerning Koszul and G-quadratic properties of algebras associated with points, curves, cubics and spaces of quadrics of low codimension.
In this paper, we construct a canonical linear basis for free commutative integro-differential algebras by applying the method of Gröbner-Shirshov bases. We establish the Composition-Diamond Lemma for free commutative differential Rota-Baxter algebras of order $n$. We also obtain a weakly monomial order on these algebras, allowing us to obtain Gröbner-Shirshov bases for free commutative integro-differential algebras on a set. We finally generalize the concept of functional derivations to free differential algebras with arbitrary weight and generating sets from which to construct a canonical li…
Algebraic cryptanalysis usually requires to recover the secret key by solving polynomial equations. Faugere's F4 is a well-known Grobner bases algorithm to solve this problem. However, a serious drawback exists in the Grobner bases based algebraic attacks, namely, any information won't be got if we couldn't work out the Grobner bases of the polynomial equations system. In this paper, we in-depth research the F4 algorithm over GF(2). By using S-polynomials to replace critical pairs and computing the normal form of the productions with respect to the field equations in certain steps, many "redun…
