Venkatesan Guruswami, Madhu Sudan

We present an improved list decoding algorithm for decoding

Reed-Solomon codes. Given an arbitrary string of length n, the

list decoding problem is that of finding all codewords within a

specified Hamming distance from the input string.

It is well-known that this decoding problem for Reed-Solomon

codes reduces to the ...
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Oded Goldreich, Dana Ron, Madhu Sudan

The Chinese Remainder Theorem states that a positive

integer m is uniquely specified by its remainder modulo

k relatively prime integers p_1,...,p_k, provided

m < \prod_{i=1}^k p_i. Thus the residues of m modulo

relatively prime integers p_1 < p_2 < ... < p_n

form a redundant representation of m if ...
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Piotr Indyk

Spielman showed that one can construct error-correcting codes capable

of correcting a constant fraction $\delta << 1/2$ of errors,

and that are encodable/decodable in linear time.

Guruswami and Sudan showed that it is possible to correct

more than $50\%$ of errors (and thus exceed the ``half of the ...
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Luca Trevisan

We show that Yao's XOR Lemma, and its essentially equivalent

rephrasing as a Direct Product Lemma, can be

re-interpreted as a way of obtaining error-correcting

codes with good list-decoding algorithms from error-correcting

codes having weak unique-decoding algorithms. To get codes

with good rate and efficient list decoding algorithms

one needs ...
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Venkatesan Guruswami

We present an explicit construction of codes that can be list decoded

from a fraction $(1-\eps)$ of errors in sub-exponential time and which

have rate $\eps/\log^{O(1)}(1/\eps)$. This comes close to the optimal

rate of $\Omega(\eps)$, and is the first sub-exponential complexity

construction to beat the rate of $O(\eps^2)$ achieved by ...
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Venkatesan Guruswami

This paper is concerned with a new family of error-correcting codes

based on algebraic curves over finite fields, and list decoding

algorithms for them. The basic goal in the subject of list decoding is

to construct error-correcting codes $C$ over some alphabet $\Sigma$

which have good rate $R$, and at ...
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Irit Dinur, Elena Grigorescu, Swastik Kopparty, Madhu Sudan

Given a pair of finite groups $G$ and $H$, the set of homomorphisms from $G$ to $H$ form an error-correcting code where codewords differ in at least $1/2$ the coordinates. We show that for every pair of {\em abelian} groups $G$ and $H$, the resulting code is (locally) list-decodable from ... more >>>

Dan Gutfreund, Guy Rothblum

We study the complexity of locally list-decoding binary error correcting codes with good parameters (that are polynomially related to information theoretic bounds). We show that computing majority over $\Theta(1/\eps)$ bits is essentially equivalent to locally list-decoding binary codes from relative distance $1/2-\eps$ with list size $\poly(1/\eps)$. That is, a local-decoder ... more >>>

Venkatesan Guruswami, Atri Rudra

We construct binary linear codes that are efficiently list-decodable

up to a fraction $(1/2-\eps)$ of errors. The codes encode $k$ bits

into $n = {\rm poly}(k/\eps)$ bits and are constructible and

list-decodable in time polynomial in $k$ and $1/\eps$ (in

particular, in our results $\eps$ need ...
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Russell Impagliazzo, Ragesh Jaiswal, Valentine Kabanets, Avi Wigderson

The classical Direct-Product Theorem for circuits says

that if a Boolean function $f:\{0,1\}^n\to\{0,1\}$ is somewhat hard

to compute on average by small circuits, then the corresponding

$k$-wise direct product function

$f^k(x_1,\dots,x_k)=(f(x_1),\dots,f(x_k))$ (where each

$x_i\in\{0,1\}^n$) is significantly harder to compute on average by

slightly smaller ...
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Parikshit Gopalan, Venkatesan Guruswami, Prasad Raghavendra, Prasad Raghavendra

We design the first efficient algorithms and prove new combinatorial bounds for list decoding tensor products of codes and interleaved codes.

1)We show that for every code, the ratio of its list decoding radius to its minimum distance stays unchanged under the tensor product operation (rather than squaring, as one ... more >>>

Shachar Lovett, Tali Kaufman

In this work we study the list-decoding size of Reed-Muller codes. Given a received word and a distance parameter, we are interested in bounding the size of the list of Reed-Muller codewords that are within that distance from the received word. Previous bounds of Gopalan, Klivans and Zuckerman~\cite{GKZ08} on the ... more >>>

Venkatesan Guruswami

Algebraic codes that achieve list decoding capacity were recently

constructed by a careful ``folding'' of the Reed-Solomon code. The

``low-degree'' nature of this folding operation was crucial to the list

decoding algorithm. We show how such folding schemes conducive to list

decoding arise out of the Artin-Frobenius automorphism at primes ...
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Zeev Dvir, Swastik Kopparty, Shubhangi Saraf, Madhu Sudan

We extend the ``method of multiplicities'' to get the following results, of interest in combinatorics and randomness extraction.

\begin{enumerate}

\item We show that every Kakeya set in $\F_q^n$, the $n$-dimensional vector space over the finite field on $q$ elements, must be of size at least $q^n/2^n$. This bound is tight ...
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Atri Rudra, steve uurtamo

We prove the following results concerning the list decoding of error-correcting codes:

We show that for any code with a relative distance of $\delta$

(over a large enough alphabet), the

following result holds for random errors: With high probability,

for a $\rho\le \delta -\eps$ fraction of random errors (for any ...
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Venkatesan Guruswami, Adam Smith

In this paper, we consider coding schemes for computationally bounded channels, which can introduce an arbitrary set of errors as long as (a) the fraction of errors is bounded with high probability by a parameter p and (b) the process which adds the errors can be described by a sufficiently ... more >>>

mohammad iftekhar husain, steve ko, Atri Rudra, steve uurtamo

We consider the following problem that arises in outsourced storage: a user stores her data $x$ on a remote server but wants to audit the server at some later point to make sure it actually did store $x$. The goal is to design a (randomized) verification protocol that has the ... more >>>

Elena Grigorescu, Chris Peikert

The question of list decoding error-correcting codes over finite fields (under the Hamming metric) has been widely studied in recent years. Motivated by the similar discrete structure of linear codes and point lattices in $R^{N}$, and their many shared applications across complexity theory, cryptography, and coding theory, we initiate the ... more >>>

Irit Dinur, Elazar Goldenberg

We consider the following clustering with outliers problem: Given a set of points $X \subset \{-1,1\}^n$, such that there is some point $z \in \{-1,1\}^n$ for which at least $\delta$ of the points are $\epsilon$-correlated with $z$, find $z$. We call such a point $z$ a $(\delta,\epsilon)$-center of X.

In ... more >>>

Abhishek Bhowmick, Shachar Lovett

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-Solomon or Reed-Muller codes.

Fix a finite field $\mathbb{F}$. ... more >>>

Atri Rudra, Mary Wootters

In this work, we introduce a framework to study the effect of random operations on the combinatorial list decodability of a code.

The operations we consider correspond to row and column operations on the matrix obtained from the code by stacking the codewords together as columns. This captures many natural ...
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Arnab Bhattacharyya, Abhishek Bhowmick

Higher-order Fourier analysis, developed over prime fields, has been recently used in different areas of computer science, including list decoding, algorithmic decomposition and testing. We extend the tools of higher-order Fourier analysis to analyze functions over general fields. Using these new tools, we revisit the results in the above areas.

... more >>>Abhishek Bhowmick, Shachar Lovett

Let $f$ be a polynomial of degree $d$ in $n$ variables over a finite field $\mathbb{F}$. The polynomial is said to be unbiased if the distribution of $f(x)$ for a uniform input $x \in \mathbb{F}^n$ is close to the uniform distribution over $\mathbb{F}$, and is called biased otherwise. The polynomial ... more >>>

Boris Bukh, Venkatesan Guruswami

We consider codes over fixed alphabets against worst-case symbol deletions. For any fixed $k \ge 2$, we construct a family of codes over alphabet of size $k$ with positive rate, which allow efficient recovery from a worst-case deletion fraction approaching $1-\frac{2}{k+1}$. In particular, for binary codes, we are able to ... more >>>

Venkatesan Guruswami, Nicolas Resch, Chaoping Xing

For a vector space $\mathbb{F}^n$ over a field $\mathbb{F}$, an $(\eta,\beta)$-dimension expander of degree $d$ is a collection of $d$ linear maps $\Gamma_j : \mathbb{F}^n \to \mathbb{F}^n$ such that for every subspace $U$ of $\mathbb{F}^n$ of dimension at most $\eta n$, the image of $U$ under all the maps, $\sum_{j=1}^d ... more >>>

Avraham Ben-Aroya, Dean Doron, Amnon Ta-Shma

A code $\mathcal{C}$ is $(1-\tau,L)$ erasure list-decodable if for every codeword $w$, after erasing any $1-\tau$ fraction of the symbols of $w$,

the remaining $\tau$-fraction of its symbols have at most $L$ possible completions into codewords of $\mathcal{C}$.

Non-explicitly, there exist binary $(1-\tau,L)$ erasure list-decodable codes having rate $O(\tau)$ and ...
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