Related papers: List Decoding of Reed-Solomon Codes from a Groebne…

In a recent paper, Brakensiek, Gopi and Makam introduced higher order MDS codes as a generalization of MDS codes. An order-$\ell$ MDS code, denoted by $\operatorname{MDS}(\ell)$, has the property that any $\ell$ subspaces formed from…

Given a classical error-correcting block code, the task of quantum list decoding is to produce from any quantumly corrupted codeword a short list containing all messages whose codewords exhibit high "presence" in the quantumly corrupted…

Algebraic codes that achieve list decoding capacity were recently constructed by a careful ``folding'' of the Reed-Solomon code. The ``low-degree'' nature of this folding operation was crucial to the list decoding algorithm. We show how…

We present an algorithm for computing Groebner bases of vanishing ideals of points that is optimized for the case when the number of points in the associated variety is less than the number of indeterminates. The algorithm first identifies…

The list decoding problem for a code asks for the maximal radius up to which any ball of that radius contains only a constant number of codewords. The list decoding radius is not well understood even for well studied codes, like…

Let T(x) in k[x] be a monic non-constant polynomial and write R=k[x] / (T) the quotient ring. Consider two bivariate polynomials a(x, y), b(x, y) in R[y]. In a first part, T = p^e is assumed to be the power of an irreducible polynomial p. A…

We show that polynomial codes (and some related codes) used for distributed matrix multiplication are interleaved Reed-Solomon codes and, hence, can be collaboratively decoded. We consider a fault tolerant setup where $t$ worker nodes…

In this paper show that the list and bounded-distance decoding problems of certain bounds for the Reed-Solomon code are at least as hard as the discrete logarithm problem over finite fields.

The concept of prefactors is considered in order to decrease the complexity of the Guruswami-Sudan interpolation step for generalized Reed-Solomon codes. It is shown that the well-known re-encoding projection due to Koetter et al. leads to…

In this paper, a new approach for decoding low-rate Reed-Solomon codes beyond half the minimum distance is considered and analyzed. Unlike the Sudan algorithm published in 1997, this new approach is based on multi-sequence shift-register…

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…

We define a new type of ideal basis called the proper basis that improves both Gr\"obner basis and Buchberger's algorithm. Let $x_1$ be the least variable of a monomial ordering in a polynomial ring $K[x_1,\dotsc,x_n]$ over a field $K$. The…

We develop the theory of Gr\"obner bases for ideals in a polynomial ring with countably infinite variables over a field. As an application we reconstruct some of the one-one correspondences among various sets of partitions by using division…

Explicit bases for the subfield subcodes of projective Reed-Muller codes over the projective plane and their duals are obtained. In particular, we provide a formula for the dimension of these codes. For the general case over the projective…

We introduce Reed-Solomon-Gabidulin codes which is, at the same time, an extension to Reed-Solomon codes on the one hand and Gabidulin codes on the other hand. We prove that our codes have good properties with respect to the minimal…

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…

Polynomial remainder codes are a large class of codes derived from the Chinese remainder theorem that includes Reed-Solomon codes as a special case. In this paper, we revisit these codes and study them more carefully than in previous work.…

We present a new upper bound for the orders of derivatives in the Rosenfeld-Groebner algorithm. This algorithm computes a regular decomposition of a radical differential ideal in the ring of differential polynomials over a differential…

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…

Reed-Muller codes encode an $m$-variate polynomial of degree $r$ by evaluating it on all points in $\{0,1\}^m$. We denote this code by $RM(m,r)$. The minimal distance of $RM(m,r)$ is $2^{m-r}$ and so it cannot correct more than half that…