Related papers: Decoding Algorithms for Twisted GRS Codes
Maximum distance separable (MDS) codes are optimal in the sense that the minimum distance cannot be improved for a given length and code size. The most prominent MDS codes are generalized Reed-Solomon (GRS) codes. The square…
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…
A complexity-adaptive tree search algorithm is proposed for $\boldsymbol{G}_N$-coset codes that implements maximum-likelihood (ML) decoding by using a successive decoding schedule. The average complexity is close to that of the successive…
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…
Maximum distance separable (MDS) codes are widely used in distributed storage, but naively repairing a single failure in an $(n,k)$ MDS code requires downloading the full contents of $k$ surviving nodes. Minimum storage regenerating (MSR)…
We propose a new partial decoding algorithm for $h$-interleaved one-point Hermitian codes that can decode-under certain assumptions-an error of relative weight up to $1-(\tfrac{k+g}{n})^{\frac{h}{h+1}}$, where $k$ is the dimension, $n$ the…
A recent work of Goyal, Harsha, Kumar and Shankar gave nearly linear time algorithms for the list decoding of Folded Reed-Solomon codes (FRS) and univariate multiplicity codes up to list decoding capacity in their natural setting of…
Decoding algorithms for Reed--Solomon (RS) codes are of great interest for both practical and theoretical reasons. In this paper, an efficient algorithm, called the modular approach (MA), is devised for solving the Welch--Berlekamp (WB) key…
We consider the decoding of linear and array codes from errors when we are only allowed to download a part of the codeword. More specifically, suppose that we have encoded $k$ data symbols using an $(n,k)$ code with code length $n$ and…
Maximum-distance-separable (MDS) codes are a class of erasure codes that are widely adopted to enhance the reliability of distributed storage systems (DSS). In (n, k) MDS coded DSS, the original data are stored into n distributed nodes in…
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 extended Euclidean algorithm (EEA) for polynomial greatest common divisors is commonly used in solving the key equation in the decoding of Reed-Solomon (RS) codes, and more generally in BCH decoding. For this particular application, the…
In coding theory, Reed-Solomon codes are one of the most well-known and widely used classes of error-correcting codes. In this thesis we study and compare two major strategies known for their decoding procedure, the…
Reed-Muller codes are some of the oldest and most widely studied error-correcting codes, of interest for both their algebraic structure as well as their many algorithmic properties. A recent beautiful result of Saptharishi, Shpilka and Volk…
We consider a list decoding algorithm recently proposed by Pellikaan-Wu \cite{PW2005} for $q$-ary Reed-Muller codes $\mathcal{RM}_q(\ell, m, n)$ of length $n \leq q^m$ when $\ell \leq q$. A simple and easily accessible correctness proof is…
In 2009, a lossless compression algorithm based on 1D chaotic maps known as Generalized Lur\"{o}th Series (or GLS) has been proposed. This algorithm (GLS-coding) encodes the input message as a symbolic sequence on an appropriate 1D chaotic…
Generalized Expanded-Blaum-Roth (GEBR) codes [1] are designed for large-scale distributed storage systems that have larger recoverability for single-symbol failures, multi-column failures and multi-row failures, compared with locally…
We present a new decoding algorithm based on error locating pairs and correcting an amount of errors exceeding half the minimum distance. When applied to Reed--Solomon or algebraic geometry codes, the algorithm is a reformulation of the…
Gradient coding is a technique for straggler mitigation in distributed learning. In this paper we design novel gradient codes using tools from classical coding theory, namely, cyclic MDS codes, which compare favorably with existing…
In this paper, we consider the convertible codes with the maximum distance separable (MDS) property, which can adjust the code rate according to the failure rates of devices. We first extend the notion of convertible codes to allow initial…