Related papers: Deep Holes of Projective Reed-Solomon Codes
List recovery of error-correcting codes has emerged as a fundamental notion with broad applications across coding theory and theoretical computer science. Folded Reed-Solomon (FRS) and univariate multiplicity codes are explicit…
We provide an algebraic description for sum-rank metric codes, as quotient space of a skew polynomial ring. This approach generalizes at the same time the skew group algebra setting for rank-metric codes and the polynomial setting for codes…
A collection of sets satisfies a $(\delta,\varepsilon)$-proximity gap with respect to some property if for every set in the collection, either (i) all members of the set are $\delta$-close to the property in (relative) Hamming distance, or…
Lifted Reed-Solomon and multiplicity codes are classes of codes, constructed from specific sets of $m$-variate polynomials. These codes allow for the design of high-rate codes that can recover every codeword or information symbol from many…
We propose a new class of efficient decoding algorithms for Reed-Muller (RM) codes over binary-input memoryless channels. The algorithms are based on projecting the code on its cosets, recursively decoding the projected codes (which are…
We construct maximally recoverable codes (corresponding to partial MDS codes) which are based on linearized Reed-Solomon codes. The new codes have a smaller field size requirement compared with known constructions. For certain asymptotic…
List-decodability of Reed-Solomon codes has received a lot of attention, but the best-possible dependence between the parameters is still not well-understood. In this work, we focus on the case where the list-decoding radius is of the form…
Constructing locally repairable codes achieving Singleton-type bound (we call them optimal codes in this paper) is a challenging task and has attracted great attention in the last few years. Tamo and Barg \cite{TB14} first gave a…
In this paper we address the problem of decoding linearized Reed-Solomon (LRS) codes beyond their unique decoding radius. We analyze the complexity in order to evaluate if the considered problem is of cryptographic relevance, i.e., can be…
A linear code with parameters of the form $[n, k, n-k+1]$ is referred to as an MDS (maximum distance separable) code. A linear code with parameters of the form $[n, k, n-k]$ is said to be almost MDS (i.e., almost maximum distance separable)…
The projective degrees of strict partitions of n were computed for all n < 101 and the partitions with maximal projective degree were found for each n. It was observed that maximizing partitions for successive values of n "lie close to each…
In this paper, we present three new classes of $q$-ary quantum MDS codes utilizing generalized Reed-Solomon codes satisfying Hermitian self-orthogonal property. Among our constructions, the minimum distance of some $q$-ary quantum MDS codes…
Let $q=p^h$ be a prime power and $e$ be an integer with $0\leq e\leq h-1$. $e$-Galois self-orthogonal codes are generalizations of Euclidean self-orthogonal codes ($e=0$) and Hermitian self-orthogonal codes ($e=\frac{h}{2}$ and $h$ is…
We present a generalisation of Twisted Reed-Solomon codes containing a new large class of MDS codes. We prove that the code class contains a large subfamily that is closed under duality. Furthermore, we study the Schur squares of the new…
This paper contributes to maximum distance separable (MDS) and near MDS (NMDS) properties of the extended generalized twisted Reed-Solomon (TGRS) codes. Firstly, a family of extended TGRS (ETGRS) are constructed by appending three columns…
One of the central tasks in quantum error-correction is to construct quantum codes that have good parameters. In this paper, we construct three new classes of quantum MDS codes from classical Hermitian self-orthogonal generalized…
In this paper we provide a detailed description of Reed-Solomon (RS) codes, the most important algorithms for decoding them, and their use in concatenated coding systems for space applications. In the current literature there is scattered…
Recursive list decoding of Reed-Muller (RM) codes, with moderate list size, is known to approach maximum-likelihood (ML) performance of short length $(\leq 256)$ RM codes. Recursive decoding employs the Plotkin construction to split the…
We introduce concepts of "recursive polynomial remainder sequence (PRS)" and "recursive subresultant," along with investigation of their properties. A recursive PRS is defined as, if there exists the GCD (greatest common divisor) of initial…
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…