Related papers: Decoding of Interleaved Reed-Solomon Codes Using I…
It is well known that a random q-ary code of rate \Omega(\epsilon^2) is list decodable up to radius (1 - 1/q - \epsilon) with list sizes on the order of 1/\epsilon^2, with probability 1 - o(1). However, until recently, a similar statement…
For communication systems with heavy burst noise, an optimal Forward Error Correction (FEC) scheme is expected to have a large burst error correction capacity while simultaneously owning moderate random error correction capability. This…
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…
In this paper, we prove that explicit FRS codes and multiplicity codes achieve relaxed generalized Singleton bounds for list size $L\ge1.$ Specifically, we show the following: (1) FRS code of length $n$ and rate $R$ over the alphabet…
We design the first efficient algorithms and prove new combinatorial bounds for list decoding tensor products of codes and interleaved codes. We show that for {\em every} code, the ratio of its list decoding radius to its minimum distance…
We present a novel iterative decoding algorithm for Reed-Muller (RM) codes, which takes advantage of a graph representation of the code. Vertices of the considered graph correspond to codewords, with two vertices being connected by an edge…
We present algorithms performing sparse univariate polynomial interpolation with errors in the evaluations of the polynomial. Based on the initial work by Comer, Kaltofen and Pernet [Proc. ISSAC 2012], we define the sparse polynomial…
We prove the following results concerning the list decoding of error-correcting codes: (i) We show that for \textit{any} code with a relative distance of $\delta$ (over a large enough alphabet), the following result holds for \textit{random…
We examine regular and irregular repeat-accumulate (RA) codes with repetition degrees which are all even. For these codes and with a particular choice of an interleaver, we give an upper bound on the decoding error probability of a…
In this paper we consider a Metzner-Kapturowski-like decoding algorithm for high-order interleaved sum-rank-metric codes, offering a novel perspective on the decoding process through the concept of an error code. The error code, defined as…
We analyze the list-decodability, and related notions, of random linear codes. This has been studied extensively before: there are many different parameter regimes and many different variants. Previous works have used complementary styles…
We explore the relationship between polar and RM codes and we describe a coding scheme which improves upon the performance of the standard polar code at practical block lengths. Our starting point is the experimental observation that RM…
This paper studies the parameters for which Reed-Muller (RM) codes over $GF(2)$ can correct random erasures and random errors with high probability, and in particular when can they achieve capacity for these two classical channels.…
In this work, we study the performance of Reed--Solomon codes against adversarial insertion-deletion (insdel) errors. We prove that over fields of size $n^{O(k)}$ there are $[n,k]$ Reed-Solomon codes that can decode from $n-2k+1$ insdel…
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…
We propose new repair schemes for Reed-Solomon codes that use subspace polynomials and hence generalize previous works in the literature that employ trace polynomials. The Reed-Solomon codes are over $\mathbb{F}_{q^\ell}$ and have…
We prove that a random linear code over F_q, with probability arbitrarily close to 1, is list decodable at radius (1-1/q-\epsilon) with list size L=O(1/\epsilon^2) and rate R=\Omega_q(\epsilon^2/(log^3(1/\epsilon))). Up to 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…
We initiate a study of locally decodable codes with randomized encoding. Standard locally decodable codes are error correcting codes with a deterministic encoding function and a randomized decoding function, such that any desired message…
The maximum-likelihood decoding problem is known to be NP-hard for general linear and Reed-Solomon codes. In this paper, we introduce the notion of A-covered codes, that is, codes that can be decoded through a polynomial time algorithm A…