Related papers: A decoding algorithm for binary linear codes using…
We present an algorithm for computing the set of all coset leaders of a binary code $\mathcal C \subset \mathbb{F}_2^n$. The method is adapted from some of the techniques related to the computation of Gr\"obner representations associated…
We present a construction of subspace codes along with an efficient algorithm for list decoding from both insertions and deletions, handling an information-theoretically maximum fraction of these with polynomially small rate. Our…
We propose a new more efficient method for the computation of two-sided Gr\"obner bases of ideals and bimodules shifting the problem to the enveloping algebra. Arising from the ideas this method involves, we introduce the notion of…
In this work we develop the theory of Gr\"obner bases for modules over the ring of univariate linearized polynomials with coefficients from a finite field.
This paper addresses the problem of learning binary hash codes for large scale image search by proposing a novel hashing method based on deep neural network. The advantage of our deep model over previous deep model used in hashing is that…
This extended abstract gives a construction for lifting a Gr\"obner basis algorithm for an ideal in a polynomial ring over a commutative ring R under the condition that R also admits a Gr\"obner basis for every ideal in R.
Developed by Buchberger for commutative polynomial rings, Groebner Bases are frequently applied to solve algorithmic problems, such as the congruence problem for ideals. Until now, these ideas have been transmitted to different in part…
The objective of this paper is to construct a class of linear codes with two nonzero weights and three nonzero weights by using the general trace functions, which weight distributions has been determined. These linear codes contain some…
The algebraic method provides useful techniques to identify models in designs and to understand aliasing of polynomial models. The present note surveys the topic of Gr\"obner bases in experimental design and then describes the notion of…
An alternative method for collaborative decoding of interleaved Reed-Solomon codes as well as Gabidulin codes for the case of high interleaving degree is proposed. As an example of application, simulation results are presented for a…
We present Binomials, a package for the computer algebra system Macaulay2, which specializes well known algorithms to binomial ideals. These come up frequently in algebraic statistics and commutative algebra, and it is shown that…
We present a novel technique for encoding and decoding constant weight binary codes that uses a geometric interpretation of the codebook. Our technique is based on embedding the codebook in a Euclidean space of dimension equal to the weight…
Prime-based ordering which is proved to be admissible, is the encoding of indeterminates in power-products with prime numbers and ordering them by using the natural number order. Using Eiffel, four versions of Buchberger's improved…
We show how Gabidulin codes can be list decoded by using a parametrization approach. For this we consider a certain module in the ring of linearized polynomials and find a minimal basis for this module using the Euclidean algorithm with…
We give new proofs of asymptotic upper bounds of coding theory obtained within the frame of Delsarte's linear programming method. The proofs rely on the analysis of eigenvectors of some finite-dimensional operators related to orthogonal…
A statistical estimation algorithm of the weight distribution of a linear code is shown, based on using its generator matrix as a compression function on random bit strings.
In this work, we introduce a framework to study the effect of random operations on the combinatorial list-decodability of a code. The operations we consider correspond to row and column operations on the matrix obtained from the code by…
In this paper, binomial difference ideals are studied. Three canonical representations for Laurent binomial difference ideals are given in terms of the reduced Groebner basis of Z[x]-lattices, regular and coherent difference ascending…
In [4] we describe a variation of the classical permutation decoding algorithm that can be applied to any binary affine-invariant code; in particular, it can be applied to first-order Reed-Muller codes successfully. In this paper we study…
In this article we present a parallel modular algorithm to compute all solutions with multiplicities of a given zero-dimensional polynomial system of equations over the rationals. In fact, we compute a triangular decomposition using…