Related papers: Generalized GM-MDS: Polynomial Codes are Higher Or…
The classical family of Reed-Solomon codes consist of evaluations of polynomials over the finite field $\mathbb{F}_q$ of degree less than $k$, at $n$ distinct field elements. These are arguably the most widely used and studied codes, as…
Generalized Reed-Solomon (GRS) and Gabidulin codes have been proposed for various code-based cryptosystems, though most such schemes without elaborate disguising techniques have been successfully attacked. Both code classes are prominent…
This paper shows that, with high probability, randomly punctured Reed-Solomon codes over fields of polynomial size achieve the list decoding capacity. More specifically, we prove that for any $\epsilon>0$ and $R\in (0,1)$, with high…
Maximum distance separable convolutional codes are the codes that present best performance in error correction among all convolutional codes with certain rate and degree. In this paper, we show that taking the constant matrix coefficients…
We investigate two classes of extended codes and provide necessary and sufficient conditions for these codes to be non-GRS MDS codes. We also determine the parity check matrices for these codes. Using the connection of MDS codes with arcs…
This paper shows that there exist Reed--Solomon (RS) codes, over \black{exponentially} large finite fields \black{in the code length}, that are combinatorially list-decodable well beyond the Johnson radius, in fact almost achieving the…
Maximum rank distance codes denoted MRD-codes are the equivalent in rank metric of MDS-codes. Given any integer $q$ power of a prime and any integer $n$ there is a family of MRD-codes of length $n$ over $\FF{q^n}$ having polynomial-time…
Maximum distance separable (MDS) and almost maximum distance separable (AMDS) codes have been widely used in various fields such as communication systems, data storage, and quantum codes because of their algebraic properties and excellent…
The notion of universally decodable matrices (UDMs) was recently introduced by Tavildar and Viswanath while studying slow fading channels. It turns out that the problem of constructing UDMs is tightly connected to the problem of…
Motivated by a problem in computer architecture we introduce a notion of the perfect distance-dominating set, PDDS, in a graph. PDDSs constitute a generalization of perfect Lee codes, diameter perfect codes, as well as other codes and…
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…
The parameters of a $q$-ary MDS Euclidean self-dual codes are completely determined by its length and the construction of MDS Euclidean self-dual codes with new length has been widely investigated in recent years. In this paper, we give a…
We describe a new class of list decodable codes based on Galois extensions of function fields and present a list decoding algorithm. These codes are obtained as a result of folding the set of rational places of a function field using…
The sum-rank metric is the mixture of the Hamming and rank metrics. The sum-rank metric found its application in network coding, locally repairable codes, space-time coding, and quantum-resistant cryptography. Linearized Reed-Solomon (LRS)…
Both MDS and Euclidean self-dual codes have theoretical and practical importance and the study of MDS self-dual codes has attracted lots of attention in recent years. In particular, determining existence of $q$-ary MDS self-dual codes for…
We study linear codes that maximize minimum distance subject to arbitrary support constraints on the parity-check matrix. Such constraints arise naturally in the design of LDPC codes, locally repairable codes, and hardware-constrained…
In this work, we present an abstract framework for some algebraic error-correcting codes with the aim of capturing codes that are list-decodable to capacity, along with their decoding algorithm. In the polynomial ideal framework, a code is…
New families of maximum distance separable (MDS) codes are constructed from elliptic curves by exploiting their group structures. In contrast to classical constructions based on divisors supported at a single rational point, the proposed…
Folded Reed-Solomon (FRS) and univariate multiplicity codes are prominent polynomial codes over finite fields, renowned for achieving list decoding capacity. These codes have found a wide range of applications beyond the traditional scope…
A general class of polynomial remainder codes is considered. Such codes are very flexible in rate and length and include Reed-Solomon codes as a special case. As an extension of previous work, two joint error-and-erasure decoding approaches…