Related papers: Decoding Reed-Solomon codes by solving a bilinear …
Many computer vision applications require robust estimation of the underlying geometry, in terms of camera motion and 3D structure of the scene. These robust methods often rely on running minimal solvers in a RANSAC framework. In this paper…
Several problems in algebraic geometry and coding theory over finite rings are modeled by systems of algebraic equations. Among these problems, we have the rank decoding problem, which is used in the construction of public-key cryptography.…
In this paper, a lemma in algebraic coding theory is established, which is frequently appeared in the encoding and decoding for algebraic codes such as Reed-Solomon codes and algebraic geometry codes. This lemma states that two vector…
We are concerned with the problem of decomposing the parameter space of a parametric system of polynomial equations, and possibly some polynomial inequality constraints, with respect to the number of real solutions that the system attains.…
In this paper we present two algorithms for the computation of a diagonal form of a matrix over non-commutative Euclidean domain over a field with the help of Gr\"obner bases. This can be viewed as the pre-processing for the computation of…
Modern spacecraft communication systems rely on concatenated error correction schemes, typically combining convolutional and Reed-Solomon (RS) codes. This paper presents a decoder-side method that uses a machine learning model to estimate…
An efficient evaluation method is described for polynomials in finite fields. Its complexity is shown to be lower than that of standard techniques when the degree of the polynomial is large enough. Applications to the syndrome computation…
We show that decoding of $\ell$-Interleaved Gabidulin codes, as well as list-$\ell$ decoding of Mahdavifar--Vardy codes can be performed by row reducing skew polynomial matrices. Inspired by row reduction of $\F[x]$ matrices, we develop a…
We investigate the list decodability of symbol-pair codes in the present paper. Firstly, we show that list decodability of every symbol-pair code does not exceed the Gilbert-Varshamov bound. On the other hand, we are able to prove that with…
We reformulate a recently introduced interpolation-based unique decoding algorithm of algebraic geometry codes using the theory of Gr\"obner bases of modules on the coordinate ring of the base curve. With the same decoding performance, the…
Solving a polynomial system, or computing an associated Gr\"obner basis, has been a fundamental task in computational algebra. However, it is also known for its notorious doubly exponential time complexity in the number of variables in the…
We consider recursive decoding for Reed-Muller (RM) codes and their subcodes. Two new recursive techniques are described. We analyze asymptotic properties of these algorithms and show that they substantially outperform other decoding…
Reed-Muller (RM) and polar codes are a class of capacity-achieving channel coding schemes with the same factor graph representation. Low-complexity decoding algorithms fall short in providing a good error-correction performance for RM and…
We define a virtual projection of a Reed-Solomon code $RS(q^{l},n,k)$ to an $RS(q,n,k)$ Reed-Solomon code. A new probabilistic decoding algorithm that can be used to perform fractional decoding beyond the $\alpha$- decoding radius is…
This article discusses the decoding of Gabidulin codes and shows how to extend the usual decoder to any supercode of a Gabidulin code at the cost of a significant decrease of the decoding radius. Using this decoder, we provide polynomial…
Despite their exceptional error-correcting properties, Reed-Solomon (RS) codes have been overlooked in distributed storage applications due to the common belief that they have poor repair bandwidth: A naive repair approach would require the…
A simple and natural Gao algorithm for decoding algebraic codes is described. Its relation to the Welch-Berlekamp and Euclidean algorithms is given.
A novel permutation decoding method for Reed-Muller codes is presented. The complexity and the error correction performance of the suggested permutation decoding approach are similar to that of the recursive lists decoder. It is…
In a recent breakthrough [BGM23, GZ23, AGL23], it was shown that randomly punctured Reed-Solomon codes are list decodable with optimal list size with high probability, i.e., they attain the Singleton bound for list decoding [ST20, Rot22,…
List decoding of codes can be seen as the generalization of unique decoding of codes While list decoding over finite fields has been extensively studied, extending these results to more general algebraic structures such as Galois rings…