On the ideal associated to a linear code (original) (raw)
Gröbner bases and combinatorics for binary codes
Applicable Algebra in Engineering, Communication and Computing, 2008
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. In the second one, 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.
Groebner bases and combinatorics for binary codes
2005
In this paper we introduce a binomial ideal derived from a binary linear code. We present some applications of a Gr\"obner 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
On a Grobner bases structure associated to linear codes
2005
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.
A decoding algorithm for binary linear codes using Groebner bases
arXiv (Cornell University), 2018
It has been discovered that linear codes may be described by binomial ideals. This makes it possible to study linear codes by commutative algebra and algebraic geometry methods. In this paper, we give a decoding algorithm for binary linear codes by utilizing the Groebner bases of the associated ideals.
Information and Control, 1972
Given an integer m which is a product of distinct primes Pi, a method is given for constructing codes over the ring of integers modulo m from cyclic codes over GF(pi). Specifically, if we are given a cyclic (n, ki) code over GF(pt) with minimum Hamming distance di, for each i, then we construct a code of block length n over the integers modulo m with 1-[~ p~i codewords, which is both linear and cyclic and has minimum Hamming distance mini di. i j k
On the structure of monomial codes and their generalizations
Journal of algebra combinatorics discrete structures and applications, 2023
In this paper, we are interested in monomial codes with associated vector a = (a0, a1,. .. , an−1), introduced in [4], and more generally in linear codes invariant under a monomial matrix M = diag(a0, a1,. .. , an−1)Pσ where σ is a permutation and Pσ its associated permutation matrix. We discuss some connections between monomial codes and codes invariant under an arbitrary monomial matrix M. Next, we identify monomial codes with associated vector a = (a0, a2,. .. , an−1) by the ideals of the polynomial ring R q,n := F q [x] x n − n−1 i=0 ai , via a special isomorphism ϕ a which preserves the Hamming weight and differs from the classical isomorphism used in the case of cyclic codes and their generalizations. This correspondence leads to some basic characterizations of monomial codes such as generator polynomials, parity check polynomials, and others. Next, we focus on the structure of −quasi-monomial (−QM) codes of length n = m , where on the one hand, we characterize them by the R q,m −submodules of R q,m. On the other hand, −QM codes are seen as additive monomial codes over the extension F q /F q. So, as in the case of quasi-cyclic codes [8], we characterize those codes that have F q −linear images with respect to a basis of the extension F q /F q , based on the CRT decomposition. Finally, we show that −QM codes and additive monomial codes are asymptotically good.
Algebraic methods for parameterized codes and invariants of vanishing ideals over finite fields
Finite Fields and Their Applications, 2011
Let K = Fq be a finite field with q elements and let X be a subset of a projective space P s−1 , over the field K, parameterized by Laurent monomials. Let I(X) be the vanishing ideal of X. Some of the main contributions of this paper are in determining the structure of I(X) to compute some of its invariants. It is shown that I(X) is a lattice ideal. We introduce the notion of a parameterized code arising from X and present algebraic methods to compute and study its dimension, length and minimum distance. For a parameterized code, arising from a connected graph, we are able to compute its length and to make our results more precise. If the graph is non-bipartite, we show an upper bound for the minimum distance.
The Weight Enumerator for a Class of Linear Codes
Recently, linear codes constructed from defining sets have been studied widely since they have many applications in cryptography and communication systems. In this paper, we consider a defining set where for a positive integer and an odd prime , and is the absolute trace function from onto. Define a class of-ary linear codes by where We compute the weight enumerators of the punctured codes .