Related papers: Efficiently list-decodable punctured Reed-Muller c…
Reed-Muller codes encode an $m$-variate polynomial of degree $r$ by evaluating it on all points in $\{0,1\}^m$. We denote this code by $RM(m,r)$. The minimal distance of $RM(m,r)$ is $2^{m-r}$ and so it cannot correct more than half that…
We show that any q-ary code with sufficiently good distance can be randomly punctured to obtain, with high probability, a code that is list decodable up to radius $1 - 1/q - \epsilon$ with near-optimal rate and list sizes. Our results imply…
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…
We present error-correcting codes that achieve the information-theoretically best possible trade-off between the rate and error-correction radius. Specifically, for every $0 < R < 1$ and $\eps> 0$, we present an explicit construction of…
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…
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…
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.…
The classical Reed-Muller codes over a finite field $\mathbb{F}_q$ are based on evaluations of $m$-variate polynomials of degree at most $d$ over a product set $U^m$, for some $d$ less than $|U|$. Because of their good distance properties,…
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,…
Understanding the limits of list-decoding and list-recovery of Reed-Solomon (RS) codes is of prime interest in coding theory and has attracted a lot of attention in recent decades. However, the best possible parameters for these problems…
The list decoding problem for a code asks for the maximal radius up to which any ball of that radius contains only a constant number of codewords. The list decoding radius is not well understood even for well studied codes, like…
Reed-Muller codes are among the most important classes of locally correctable codes. Currently local decoding of Reed-Muller codes is based on decoding on lines or quadratic curves to recover one single coordinate. To recover multiple…
First- and second-order Reed-Muller (RM(1) and RM(2), respectively) codes are two fundamental error-correcting codes which arise in communication as well as in probabilistically-checkable proofs and learning. In this paper, we take the…
A novel recursive list decoding (RLD) algorithm for Reed-Muller (RM) codes based on successive permutations (SP) of the codeword is presented. A low-complexity SP scheme applied to a subset of the symmetry group of RM codes is first…
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…
Folded Reed-Solomon codes are an explicit family of codes that achieve the optimal trade-off between rate and error-correction capability: specifically, for any $\eps > 0$, the author and Rudra (2006,08) presented an $n^{O(1/\eps)}$ time…
Recursive list decoding is considered for Reed-Muller (RM) codes. The algorithm repeatedly relegates itself to the shorter RM codes by recalculating the posterior probabilities of their symbols. Intermediate decodings are only performed…
Reed-Muller codes consist of evaluations of $n$-variate polynomials over a finite field $\mathbb{F}$ with degree at most $d$. Much like every linear code, Reed-Muller codes can be characterized by constraints, where a codeword is valid if…
We give a polynomial time algorithm to decode multivariate polynomial codes of degree $d$ up to half their minimum distance, when the evaluation points are an arbitrary product set $S^m$, for every $d < |S|$. Previously known algorithms can…
The classical family of $[n,k]_q$ Reed-Solomon codes over a field $\F_q$ consist of the evaluations of polynomials $f \in \F_q[X]$ of degree $< k$ at $n$ distinct field elements. In this work, we consider a closely related family of codes,…