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Electronic Colloquium on Computational Complexity

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All reports by Author Swastik Kopparty:

TR20-083 | 30th May 2020
Eli Ben-Sasson, Dan Carmon, Yuval Ishai, Swastik Kopparty, Shubhangi Saraf

Proximity Gaps for Reed-Solomon Codes

Revisions: 1

A collection of sets displays a proximity gap with respect to some property if for every set in the collection, either (i) all members are $\delta$-close to the property in relative Hamming distance or (ii) only a tiny fraction of members are $\delta$-close to the property. In particular, no set ... more >>>

TR20-029 | 6th March 2020
Swastik Kopparty, Guy Moshkovitz, Jeroen Zuiddam

Geometric Rank of Tensors and Subrank of Matrix Multiplication

Motivated by problems in algebraic complexity theory (e.g., matrix multiplication) and extremal combinatorics (e.g., the cap set problem and the sunflower problem), we introduce the geometric rank as a new tool in the study of tensors and hypergraphs. We prove that the geometric rank is an upper bound on the ... more >>>

TR19-090 | 27th June 2019
Ronen Shaltiel, Swastik Kopparty, Jad Silbak

Quasilinear time list-decodable codes for space bounded channels

Revisions: 2

We consider codes for space bounded channels. This is a model for communication under noise that was studied by Guruswami and Smith (J. ACM 2016) and lies between the Shannon (random) and Hamming (adversarial) models. In this model, a channel is a space bounded procedure that reads the codeword in ... more >>>

TR18-081 | 20th April 2018
Abhishek Bhrushundi, Prahladh Harsha, Pooya Hatami, Swastik Kopparty, Mrinal Kumar

On Multilinear Forms: Bias, Correlation, and Tensor Rank

Revisions: 1

In this paper, we prove new relations between the bias of multilinear forms, the correlation between multilinear forms and lower degree polynomials, and the rank of tensors over $GF(2)= \{0,1\}$. We show the following results for multilinear forms and tensors.

1. Correlation bounds : We show that a random $d$-linear ... more >>>

TR17-126 | 7th August 2017
Swastik Kopparty, Shubhangi Saraf

Local Testing and Decoding of High-Rate Error-Correcting Codes

We survey the state of the art in constructions of locally testable
codes, locally decodable codes and locally correctable codes of high rate.

more >>>

TR16-122 | 11th August 2016
Sivakanth Gopi, Swastik Kopparty, Rafael Mendes de Oliveira, Noga Ron-Zewi, Shubhangi Saraf

Locally testable and Locally correctable Codes Approaching the Gilbert-Varshamov Bound

One of the most important open problems in the theory
of error-correcting codes is to determine the
tradeoff between the rate $R$ and minimum distance $\delta$ of a binary
code. The best known tradeoff is the Gilbert-Varshamov bound,
and says that for every $\delta \in (0, 1/2)$, there are ... more >>>

TR15-068 | 21st April 2015
Swastik Kopparty, Noga Ron-Zewi, Shubhangi Saraf

High rate locally-correctable and locally-testable codes with sub-polynomial query complexity

Revisions: 3

In this work, we construct the first locally-correctable codes (LCCs), and locally-testable codes (LTCs) with constant rate, constant relative distance, and sub-polynomial query complexity. Specifically, we show that there exist binary LCCs and LTCs with block length $n$, constant rate (which can even be taken arbitrarily close to 1), constant ... more >>>

TR15-047 | 2nd April 2015
Swastik Kopparty, Mrinal Kumar, Michael Saks

Efficient indexing of necklaces and irreducible polynomials over finite fields

We study the problem of indexing irreducible polynomials over finite fields, and give the first efficient algorithm for this problem. Specifically, we show the existence of poly(n, log q)-size circuits that compute a bijection between {1, ... , |S|} and the set S of all irreducible, monic, univariate polynomials of ... more >>>

TR14-001 | 4th January 2014
Swastik Kopparty, Shubhangi Saraf, Amir Shpilka

Equivalence of Polynomial Identity Testing and Deterministic Multivariate Polynomial Factorization

In this paper we show that the problem of deterministically factoring multivariate polynomials reduces to the problem of deterministic polynomial identity testing. Specifically, we show that given an arithmetic circuit (either explicitly or via black-box access) that computes a polynomial $f(X_1,\ldots,X_n)$, the task of computing arithmetic circuits for the factors ... more >>>

TR12-044 | 22nd April 2012
Swastik Kopparty

List-Decoding Multiplicity Codes

We study the list-decodability of multiplicity codes. These codes, which are based on evaluations of high-degree polynomials and their derivatives, have rate approaching $1$ while simultaneously allowing for sublinear-time error-correction. In this paper, we show that multiplicity codes also admit powerful list-decoding and local list-decoding algorithms correcting a large fraction ... more >>>

TR10-148 | 23rd September 2010
Swastik Kopparty, Shubhangi Saraf, Sergey Yekhanin

High-rate codes with sublinear-time decoding

Locally decodable codes are error-correcting codes that admit efficient decoding algorithms; any bit of the original message can be recovered by looking at only a small number of locations of a corrupted codeword. The tradeoff between the rate of a code and the locality/efficiency of its decoding algorithms has been ... more >>>

TR10-044 | 12th March 2010
Eli Ben-Sasson, Swastik Kopparty

Affine Dispersers from Subspace Polynomials

{\em Dispersers} and {\em extractors} for affine sources of dimension $d$ in $\mathbb F_p^n$ --- where $\mathbb F_p$ denotes the finite field of prime size $p$ --- are functions $f: \mathbb F_p^n \rightarrow \mathbb F_p$ that behave pseudorandomly when their domain is restricted to any particular affine space $S \subseteq ... more >>>

TR10-003 | 6th January 2010
Venkatesan Guruswami, Johan HÃ¥stad, Swastik Kopparty

On the List-Decodability of Random Linear Codes

For every fixed finite field $\F_q$, $p \in (0,1-1/q)$ and $\varepsilon >
0$, we prove that with high probability a random subspace $C$ of
$\F_q^n$ of dimension $(1-H_q(p)-\varepsilon)n$ has the
property that every Hamming ball of radius $pn$ has at most
$O(1/\varepsilon)$ codewords.

This ... more >>>

TR09-115 | 13th November 2009
Swastik Kopparty, Shubhangi Saraf

Local list-decoding and testing of random linear codes from high-error

In this paper, we give surprisingly efficient algorithms for list-decoding and testing
{\em random} linear codes. Our main result is that random sparse linear codes are locally testable and locally list-decodable
in the {\em high-error} regime with only a {\em constant} number of queries.
More precisely, we show that ... more >>>

TR09-086 | 2nd October 2009
Arnab Bhattacharyya, Swastik Kopparty, Grant Schoenebeck, Madhu Sudan, David Zuckerman

Optimal testing of Reed-Muller codes

Revisions: 1

We consider the problem of testing if a given function
$f : \F_2^n \rightarrow \F_2$ is close to any degree $d$ polynomial
in $n$ variables, also known as the Reed-Muller testing problem.
Alon et al.~\cite{AKKLR} proposed and analyzed a natural
$2^{d+1}$-query test for this property and showed that it accepts
more >>>

TR09-033 | 16th April 2009
Phokion G. Kolaitis, Swastik Kopparty

Random Graphs and the Parity Quantifier

The classical zero-one law for first-order logic on random graphs says that for every first-order property $\varphi$ in the theory of graphs and every $p \in (0,1)$, the probability that the random graph $G(n, p)$ satisfies $\varphi$ approaches either $0$ or $1$ as $n$ approaches infinity. It is well known ... more >>>

TR09-004 | 15th January 2009
Zeev Dvir, Swastik Kopparty, Shubhangi Saraf, Madhu Sudan

Extensions to the Method of Multiplicities, with applications to Kakeya Sets and Mergers

Revisions: 2

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

TR08-020 | 7th March 2008
Irit Dinur, Elena Grigorescu, Swastik Kopparty, Madhu Sudan

Decodability of Group Homomorphisms beyond the Johnson Bound

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 >>>

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