List decoding codes on Garcia–Stictenoth tower using Gröbner basis (original) (raw)

Grobner Bases for Lattices and an Algebraic Decoding Algorithm

IEEE Transactions on Communications, 2013

In this paper we present Gröbner bases for lattices given in a general form, including integer and non-integer lattices. Gröbner bases for binary linear codes were introduced by Borges-Quintana et al. . We extend their work to nonbinary group block codes. Then, given a lattice Λ and its associated label code L, which is a group code, we define an ideal for L. A Gröbner basis is assigned to Λ as the Gröbner basis of its label code L. Since the associated label code for integer and noninteger lattices are group codes, the assigned Gröbner bases can be obtained for both cases. Using this Gröbner basis an algebraic decoding algorithm is introduced. We provide an example of the decoding method for a lower dimension lattice. We explain that the complexity of this decoding method depends on the division algorithm and show this decoding method has polynomial time complexity. Experiments for some versions of root lattices (E7 and E8) show that for low SNR the performance of these lattices is near to the lower bounds given in [16].

Gröbner Bases over Galois Rings with an Application to Decoding Alternant Codes

Journal of Symbolic Computation, 2001

We develop a theory of Gröbner bases over Galois rings, following the usual formulation for Gröbner bases over finite fields. Our treatment includes a division algorithm, a characterization of Gröbner bases, and an extension of Buchberger's algorithm. One application is towards the problem of decoding alternant codes over Galois rings. To this end we consider the module M = {(a, b) : aS ≡ b mod x r } of all solutions to the so-called key equation for alternant codes, where S is a syndrome polynomial. In decoding, a particular solution (Σ, Ω) ∈ M is sought satisfying certain conditions, and such a solution can be found in a Gröbner basis of M . Applying techniques introduced in the first part of this paper, we give an algorithm which returns the required solution.

Use of Grobner bases to decode binary cyclic codes up to the true minimum distance

IEEE Transactions on Information Theory, 1994

A general algebraic method for decoding all types of binary cyclic codes is presented. It is shown that such a method can correct t=[(d-1)/2] errors, where d is the true minimum distance of the given cyclic code. The key idea behind this decoding technique is a systematic application of the algorithmic procedures of Grobner bases to obtain the error-locator polynomial L(z). The discussion begins from a set of syndrome polynomials F and the ideal T(F) generated by F. It is proved here that the process of transforming F to the normalized reduced Grobner basis of I(F) with respect to the “purely lexicographical” ordering automatically converges to L(z). Furthermore, it is shown that L(z) can be derived from any normalized Grobner basis of I(F) with respect to any admissible total ordering. To illustrate this new approach, the procedures for decoding certain BCH codes and quadratic residue codes are demonstrated

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.

Gröbner basis approach to list decoding of algebraic geometry codes

Applicable Algebra in Engineering, Communication and Computing, 2007

We show how our Gröbner basis algorithm, which was previously applied to list decoding of Reed Solomon codes, can be used in the hard and soft decision list decoding of Algebraic Geometry codes. In addition, we present a linear functional version of our Gröbner basis algorithm in order to facilitate comparisons with methods based on duality.

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.

Computing Gröbner bases associated with lattices

Advances in Mathematics of Communications, 2016

We specialize Möller's algorithm to the computation of Gröbner bases related to lattices. We give the complexity analysis of our algorithm. Then we provide experiments showing that our algorithm is more efficient than Buchberger's algorithm for computing the associated Gröbner bases. Furthermore we show that the binomial ideal associated to the lattice can be constructed from a set of binomials associated with a set of generators of the corresponding label code. This result is presented in a general way by means of three ideal constructions associated with group codes that constitute the same ideal. This generalizes earlier results for specific cases of group codes such as linear codes, codes over Zm and label codes of lattices.

Reduced Complexity Interpolation for List Decoding Hermitian Codes

IEEE Transactions on Wireless Communications - TWC, 2008

List decoding Hermitian codes using the Guruswami-Sudan (GS) algorithm can correct errors beyond half the designed minimum distance. It consists of two processes: interpolation and factorisation. By first defining a Hermitian curve, these processes can be implemented with an iterative polynomial construction algorithm and a recursive coefficient search algorithm respectively. To improve the efficiency of list decoding Hermitian codes, this paper presents two contributions to reduce the interpolation complexity. First, in order to simplify the calculation of a polynomialiquests zero condition during the iterative interpolation, we propose an algorithm to determine the corresponding coefficients between the pole basis monomials and zero basis functions of a Hermitian curve. Second, we propose a modified complexity reducing interpolation algorithm. This scheme identifies any unnecessary polynomials during iterations and eliminates them to improve the interpolation efficiency. Due to th...

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.

A displacement approach to decoding algebraic codes

Contemporary Mathematics, 2003

Algebraic coding theory is one of the areas that routinely gives rise to computational problems involving various structured matrices, such as Hankel, Vandermonde, Cauchy matrices, and certain generalizations thereof. Their structure has often been used to derive efficient algorithms; however, the use of the structure was pattern-specific, without applying a unified technique. In contrast, in several other areas, where structured matrices are also widely encountered, the concept of displacement rank was found to be useful to derive efficient algorithms in a unified manner (i.e., not depending on a particular pattern of structure). The latter technique allows one to "compress," in a unified way, different types of n × n structured matrices to only αn parameters. This typically leads to computational savings (in many applications the number α, called the displacement rank, is a small fixed constant).