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Results for query kopparty:

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

Quasilinear time list-decodable codes for space bounded channels

Revisions: 1

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 >>>


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

Quasilinear time list-decodable codes for space bounded channels

Revisions: 1

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 >>>


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

Quasilinear time list-decodable codes for space bounded channels

Revisions: 1

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 >>>


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

Quasilinear time list-decodable codes for space bounded channels

Revisions: 1

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 >>>


TR19-080 | 1st June 2019
Swastik Kopparty, Nicolas Resch, Noga Ron-Zewi, Shubhangi Saraf, Shashwat Silas

On List Recovery of High-Rate Tensor Codes

We continue the study of list recovery properties of high-rate tensor codes, initiated by Hemenway, Ron-Zewi, and Wootters (FOCS'17). In that work it was shown that the tensor product of an efficient (poly-time) high-rate globally list recoverable code is {\em approximately} locally list recoverable, as well as globally list recoverable ... more >>>


TR19-080 | 1st June 2019
Swastik Kopparty, Nicolas Resch, Noga Ron-Zewi, Shubhangi Saraf, Shashwat Silas

On List Recovery of High-Rate Tensor Codes

We continue the study of list recovery properties of high-rate tensor codes, initiated by Hemenway, Ron-Zewi, and Wootters (FOCS'17). In that work it was shown that the tensor product of an efficient (poly-time) high-rate globally list recoverable code is {\em approximately} locally list recoverable, as well as globally list recoverable ... more >>>


TR19-080 | 1st June 2019
Swastik Kopparty, Nicolas Resch, Noga Ron-Zewi, Shubhangi Saraf, Shashwat Silas

On List Recovery of High-Rate Tensor Codes

We continue the study of list recovery properties of high-rate tensor codes, initiated by Hemenway, Ron-Zewi, and Wootters (FOCS'17). In that work it was shown that the tensor product of an efficient (poly-time) high-rate globally list recoverable code is {\em approximately} locally list recoverable, as well as globally list recoverable ... more >>>


TR19-080 | 1st June 2019
Swastik Kopparty, Nicolas Resch, Noga Ron-Zewi, Shubhangi Saraf, Shashwat Silas

On List Recovery of High-Rate Tensor Codes

We continue the study of list recovery properties of high-rate tensor codes, initiated by Hemenway, Ron-Zewi, and Wootters (FOCS'17). In that work it was shown that the tensor product of an efficient (poly-time) high-rate globally list recoverable code is {\em approximately} locally list recoverable, as well as globally list recoverable ... more >>>


TR19-080 | 1st June 2019
Swastik Kopparty, Nicolas Resch, Noga Ron-Zewi, Shubhangi Saraf, Shashwat Silas

On List Recovery of High-Rate Tensor Codes

We continue the study of list recovery properties of high-rate tensor codes, initiated by Hemenway, Ron-Zewi, and Wootters (FOCS'17). In that work it was shown that the tensor product of an efficient (poly-time) high-rate globally list recoverable code is {\em approximately} locally list recoverable, as well as globally list recoverable ... more >>>


TR19-044 | 28th March 2019
Eli Ben-Sasson, Lior Goldberg, Swastik Kopparty, Shubhangi Saraf

DEEP-FRI: Sampling Outside the Box Improves Soundness

Revisions: 2

Motivated by the quest for scalable and succinct zero knowledge arguments, we revisit worst-case-to-average-case reductions for linear spaces, raised by [Rothblum, Vadhan, Wigderson, STOC 2013]. The previous state of the art by [Ben-Sasson, Kopparty, Saraf, CCC 2018] showed that if some member of an affine space $U$ is $\delta$-far in ... more >>>


TR19-044 | 28th March 2019
Eli Ben-Sasson, Lior Goldberg, Swastik Kopparty, Shubhangi Saraf

DEEP-FRI: Sampling Outside the Box Improves Soundness

Revisions: 2

Motivated by the quest for scalable and succinct zero knowledge arguments, we revisit worst-case-to-average-case reductions for linear spaces, raised by [Rothblum, Vadhan, Wigderson, STOC 2013]. The previous state of the art by [Ben-Sasson, Kopparty, Saraf, CCC 2018] showed that if some member of an affine space $U$ is $\delta$-far in ... more >>>


TR19-044 | 28th March 2019
Eli Ben-Sasson, Lior Goldberg, Swastik Kopparty, Shubhangi Saraf

DEEP-FRI: Sampling Outside the Box Improves Soundness

Revisions: 2

Motivated by the quest for scalable and succinct zero knowledge arguments, we revisit worst-case-to-average-case reductions for linear spaces, raised by [Rothblum, Vadhan, Wigderson, STOC 2013]. The previous state of the art by [Ben-Sasson, Kopparty, Saraf, CCC 2018] showed that if some member of an affine space $U$ is $\delta$-far in ... more >>>


TR19-044 | 28th March 2019
Eli Ben-Sasson, Lior Goldberg, Swastik Kopparty, Shubhangi Saraf

DEEP-FRI: Sampling Outside the Box Improves Soundness

Revisions: 2

Motivated by the quest for scalable and succinct zero knowledge arguments, we revisit worst-case-to-average-case reductions for linear spaces, raised by [Rothblum, Vadhan, Wigderson, STOC 2013]. The previous state of the art by [Ben-Sasson, Kopparty, Saraf, CCC 2018] showed that if some member of an affine space $U$ is $\delta$-far in ... more >>>


TR19-044 | 28th March 2019
Eli Ben-Sasson, Lior Goldberg, Swastik Kopparty, Shubhangi Saraf

DEEP-FRI: Sampling Outside the Box Improves Soundness

Revisions: 2

Motivated by the quest for scalable and succinct zero knowledge arguments, we revisit worst-case-to-average-case reductions for linear spaces, raised by [Rothblum, Vadhan, Wigderson, STOC 2013]. The previous state of the art by [Ben-Sasson, Kopparty, Saraf, CCC 2018] showed that if some member of an affine space $U$ is $\delta$-far in ... more >>>


TR19-044 | 28th March 2019
Eli Ben-Sasson, Lior Goldberg, Swastik Kopparty, Shubhangi Saraf

DEEP-FRI: Sampling Outside the Box Improves Soundness

Revisions: 2

Motivated by the quest for scalable and succinct zero knowledge arguments, we revisit worst-case-to-average-case reductions for linear spaces, raised by [Rothblum, Vadhan, Wigderson, STOC 2013]. The previous state of the art by [Ben-Sasson, Kopparty, Saraf, CCC 2018] showed that if some member of an affine space $U$ is $\delta$-far in ... more >>>


TR18-091 | 4th May 2018
Swastik Kopparty, Noga Ron-Zewi, Shubhangi Saraf, Mary Wootters

Improved decoding of Folded Reed-Solomon and Multiplicity Codes

In this work, we show new and improved error-correcting properties of folded Reed-Solomon codes and multiplicity codes. Both of these families of codes are based on polynomials over finite fields, and both have been the sources of recent advances in coding theory. Folded Reed-Solomon codes were the first explicit constructions ... more >>>


TR18-091 | 4th May 2018
Swastik Kopparty, Noga Ron-Zewi, Shubhangi Saraf, Mary Wootters

Improved decoding of Folded Reed-Solomon and Multiplicity Codes

In this work, we show new and improved error-correcting properties of folded Reed-Solomon codes and multiplicity codes. Both of these families of codes are based on polynomials over finite fields, and both have been the sources of recent advances in coding theory. Folded Reed-Solomon codes were the first explicit constructions ... more >>>


TR18-091 | 4th May 2018
Swastik Kopparty, Noga Ron-Zewi, Shubhangi Saraf, Mary Wootters

Improved decoding of Folded Reed-Solomon and Multiplicity Codes

In this work, we show new and improved error-correcting properties of folded Reed-Solomon codes and multiplicity codes. Both of these families of codes are based on polynomials over finite fields, and both have been the sources of recent advances in coding theory. Folded Reed-Solomon codes were the first explicit constructions ... more >>>


TR18-091 | 4th May 2018
Swastik Kopparty, Noga Ron-Zewi, Shubhangi Saraf, Mary Wootters

Improved decoding of Folded Reed-Solomon and Multiplicity Codes

In this work, we show new and improved error-correcting properties of folded Reed-Solomon codes and multiplicity codes. Both of these families of codes are based on polynomials over finite fields, and both have been the sources of recent advances in coding theory. Folded Reed-Solomon codes were the first explicit constructions ... more >>>


TR18-090 | 4th May 2018
Eli Ben-Sasson, Swastik Kopparty, Shubhangi Saraf

Worst-case to average case reductions for the distance to a code

Revisions: 1

Algebraic proof systems reduce computational problems to problems about estimating the distance of a sequence of functions $u=(u_1,\ldots, u_k)$, given as oracles, from a linear error correcting code $V$. The soundness of such systems relies on methods that act ``locally'' on $u$ and map it to a single function $u^*$ ... more >>>


TR18-090 | 4th May 2018
Eli Ben-Sasson, Swastik Kopparty, Shubhangi Saraf

Worst-case to average case reductions for the distance to a code

Revisions: 1

Algebraic proof systems reduce computational problems to problems about estimating the distance of a sequence of functions $u=(u_1,\ldots, u_k)$, given as oracles, from a linear error correcting code $V$. The soundness of such systems relies on methods that act ``locally'' on $u$ and map it to a single function $u^*$ ... more >>>


TR18-090 | 4th May 2018
Eli Ben-Sasson, Swastik Kopparty, Shubhangi Saraf

Worst-case to average case reductions for the distance to a code

Revisions: 1

Algebraic proof systems reduce computational problems to problems about estimating the distance of a sequence of functions $u=(u_1,\ldots, u_k)$, given as oracles, from a linear error correcting code $V$. The soundness of such systems relies on methods that act ``locally'' on $u$ and map it to a single function $u^*$ ... 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 >>>


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 >>>


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 >>>


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 >>>

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 >>>

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 >>>


TR15-110 | 8th July 2015
Swastik Kopparty, Or Meir, Noga Ron-Zewi, Shubhangi Saraf

High-rate Locally-testable Codes with Quasi-polylogarithmic Query Complexity

Revisions: 1

An error correcting code is said to be \emph{locally testable} if
there is a test that checks whether a given string is a codeword,
or rather far from the code, by reading only a small number of symbols
of the string. Locally testable codes (LTCs) are both interesting
in their ... more >>>


TR15-110 | 8th July 2015
Swastik Kopparty, Or Meir, Noga Ron-Zewi, Shubhangi Saraf

High-rate Locally-testable Codes with Quasi-polylogarithmic Query Complexity

Revisions: 1

An error correcting code is said to be \emph{locally testable} if
there is a test that checks whether a given string is a codeword,
or rather far from the code, by reading only a small number of symbols
of the string. Locally testable codes (LTCs) are both interesting
in their ... more >>>


TR15-110 | 8th July 2015
Swastik Kopparty, Or Meir, Noga Ron-Zewi, Shubhangi Saraf

High-rate Locally-testable Codes with Quasi-polylogarithmic Query Complexity

Revisions: 1

An error correcting code is said to be \emph{locally testable} if
there is a test that checks whether a given string is a codeword,
or rather far from the code, by reading only a small number of symbols
of the string. Locally testable codes (LTCs) are both interesting
in their ... more >>>


TR15-110 | 8th July 2015
Swastik Kopparty, Or Meir, Noga Ron-Zewi, Shubhangi Saraf

High-rate Locally-testable Codes with Quasi-polylogarithmic Query Complexity

Revisions: 1

An error correcting code is said to be \emph{locally testable} if
there is a test that checks whether a given string is a codeword,
or rather far from the code, by reading only a small number of symbols
of the string. Locally testable codes (LTCs) are both interesting
in their ... more >>>


TR15-110 | 8th July 2015
Swastik Kopparty, Or Meir, Noga Ron-Zewi, Shubhangi Saraf

High-rate Locally-testable Codes with Quasi-polylogarithmic Query Complexity

Revisions: 1

An error correcting code is said to be \emph{locally testable} if
there is a test that checks whether a given string is a codeword,
or rather far from the code, by reading only a small number of symbols
of the string. Locally testable codes (LTCs) are both interesting
in their ... more >>>


TR15-110 | 8th July 2015
Swastik Kopparty, Or Meir, Noga Ron-Zewi, Shubhangi Saraf

High-rate Locally-testable Codes with Quasi-polylogarithmic Query Complexity

Revisions: 1

An error correcting code is said to be \emph{locally testable} if
there is a test that checks whether a given string is a codeword,
or rather far from the code, by reading only a small number of symbols
of the string. Locally testable codes (LTCs) are both interesting
in their ... 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: 2

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 >>>


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

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

Revisions: 2

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 >>>


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

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

Revisions: 2

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 >>>


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

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

Revisions: 2

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 >>>


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

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

Revisions: 2

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 >>>


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 >>>


TR14-098 | 30th July 2014
Amey Bhangale, Swastik Kopparty, Sushant Sachdeva

Simultaneous Approximation of Constraint Satisfaction Problems

Given $k$ collections of 2SAT clauses on the same set of variables $V$, can we find one assignment that satisfies a large fraction of clauses from each collection? We consider such simultaneous constraint satisfaction problems, and design the first nontrivial approximation algorithms in this context.

Our main result is that ... more >>>


TR14-098 | 30th July 2014
Amey Bhangale, Swastik Kopparty, Sushant Sachdeva

Simultaneous Approximation of Constraint Satisfaction Problems

Given $k$ collections of 2SAT clauses on the same set of variables $V$, can we find one assignment that satisfies a large fraction of clauses from each collection? We consider such simultaneous constraint satisfaction problems, and design the first nontrivial approximation algorithms in this context.

Our main result is that ... 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 >>>


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 >>>


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 >>>


TR13-085 | 13th June 2013
Eli Ben-Sasson, Yohay Kaplan, Swastik Kopparty, Or Meir, Henning Stichtenoth

Constant rate PCPs for circuit-SAT with sublinear query complexity

The PCP theorem (Arora et. al., J. ACM 45(1,3)) says that every NP-proof can be encoded to another proof, namely, a probabilistically checkable proof (PCP), which can be tested by a verifier that queries only a small part of the PCP. A natural question is how large is the blow-up ... more >>>


TR13-085 | 13th June 2013
Eli Ben-Sasson, Yohay Kaplan, Swastik Kopparty, Or Meir, Henning Stichtenoth

Constant rate PCPs for circuit-SAT with sublinear query complexity

The PCP theorem (Arora et. al., J. ACM 45(1,3)) says that every NP-proof can be encoded to another proof, namely, a probabilistically checkable proof (PCP), which can be tested by a verifier that queries only a small part of the PCP. A natural question is how large is the blow-up ... more >>>


TR13-085 | 13th June 2013
Eli Ben-Sasson, Yohay Kaplan, Swastik Kopparty, Or Meir, Henning Stichtenoth

Constant rate PCPs for circuit-SAT with sublinear query complexity

The PCP theorem (Arora et. al., J. ACM 45(1,3)) says that every NP-proof can be encoded to another proof, namely, a probabilistically checkable proof (PCP), which can be tested by a verifier that queries only a small part of the PCP. A natural question is how large is the blow-up ... more >>>


TR13-085 | 13th June 2013
Eli Ben-Sasson, Yohay Kaplan, Swastik Kopparty, Or Meir, Henning Stichtenoth

Constant rate PCPs for circuit-SAT with sublinear query complexity

The PCP theorem (Arora et. al., J. ACM 45(1,3)) says that every NP-proof can be encoded to another proof, namely, a probabilistically checkable proof (PCP), which can be tested by a verifier that queries only a small part of the PCP. A natural question is how large is the blow-up ... more >>>


TR13-060 | 10th April 2013
Venkatesan Guruswami, Swastik Kopparty

Explicit Subspace Designs

A subspace design is a collection $\{H_1,H_2,\dots,H_M\}$ of subspaces of ${\mathbf F}_q^m$ with the property that no low-dimensional subspace $W$ of ${\mathbf F}_q^m$ intersects too many subspaces of the collection. Subspace designs were introduced by Guruswami and Xing (STOC 2013) who used them to give a randomized construction of optimal ... more >>>


TR13-060 | 10th April 2013
Venkatesan Guruswami, Swastik Kopparty

Explicit Subspace Designs

A subspace design is a collection $\{H_1,H_2,\dots,H_M\}$ of subspaces of ${\mathbf F}_q^m$ with the property that no low-dimensional subspace $W$ of ${\mathbf F}_q^m$ intersects too many subspaces of the collection. Subspace designs were introduced by Guruswami and Xing (STOC 2013) who used them to give a randomized construction of optimal ... more >>>


TR13-060 | 10th April 2013
Venkatesan Guruswami, Swastik Kopparty

Explicit Subspace Designs

A subspace design is a collection $\{H_1,H_2,\dots,H_M\}$ of subspaces of ${\mathbf F}_q^m$ with the property that no low-dimensional subspace $W$ of ${\mathbf F}_q^m$ intersects too many subspaces of the collection. Subspace designs were introduced by Guruswami and Xing (STOC 2013) who used them to give a randomized construction of optimal ... more >>>


TR13-060 | 10th April 2013
Venkatesan Guruswami, Swastik Kopparty

Explicit Subspace Designs

A subspace design is a collection $\{H_1,H_2,\dots,H_M\}$ of subspaces of ${\mathbf F}_q^m$ with the property that no low-dimensional subspace $W$ of ${\mathbf F}_q^m$ intersects too many subspaces of the collection. Subspace designs were introduced by Guruswami and Xing (STOC 2013) who used them to give a randomized construction of optimal ... more >>>


TR13-060 | 10th April 2013
Venkatesan Guruswami, Swastik Kopparty

Explicit Subspace Designs

A subspace design is a collection $\{H_1,H_2,\dots,H_M\}$ of subspaces of ${\mathbf F}_q^m$ with the property that no low-dimensional subspace $W$ of ${\mathbf F}_q^m$ intersects too many subspaces of the collection. Subspace designs were introduced by Guruswami and Xing (STOC 2013) who used them to give a randomized construction of optimal ... more >>>


TR13-060 | 10th April 2013
Venkatesan Guruswami, Swastik Kopparty

Explicit Subspace Designs

A subspace design is a collection $\{H_1,H_2,\dots,H_M\}$ of subspaces of ${\mathbf F}_q^m$ with the property that no low-dimensional subspace $W$ of ${\mathbf F}_q^m$ intersects too many subspaces of the collection. Subspace designs were introduced by Guruswami and Xing (STOC 2013) who used them to give a randomized construction of optimal ... more >>>


TR12-149 | 8th November 2012
Alan Guo, Swastik Kopparty, Madhu Sudan

New affine-invariant codes from lifting

Comments: 1

In this work we explore error-correcting codes derived from
the ``lifting'' of ``affine-invariant'' codes.
Affine-invariant codes are simply linear codes whose coordinates
are a vector space over a field and which are invariant under
affine-transformations of the coordinate space. Lifting takes codes
defined over a vector space of small dimension ... more >>>


TR12-149 | 8th November 2012
Alan Guo, Swastik Kopparty, Madhu Sudan

New affine-invariant codes from lifting

Comments: 1

In this work we explore error-correcting codes derived from
the ``lifting'' of ``affine-invariant'' codes.
Affine-invariant codes are simply linear codes whose coordinates
are a vector space over a field and which are invariant under
affine-transformations of the coordinate space. Lifting takes codes
defined over a vector space of small dimension ... more >>>


TR12-149 | 8th November 2012
Alan Guo, Swastik Kopparty, Madhu Sudan

New affine-invariant codes from lifting

Comments: 1

In this work we explore error-correcting codes derived from
the ``lifting'' of ``affine-invariant'' codes.
Affine-invariant codes are simply linear codes whose coordinates
are a vector space over a field and which are invariant under
affine-transformations of the coordinate space. Lifting takes codes
defined over a vector space of small dimension ... more >>>


TR12-149 | 8th November 2012
Alan Guo, Swastik Kopparty, Madhu Sudan

New affine-invariant codes from lifting

Comments: 1

In this work we explore error-correcting codes derived from
the ``lifting'' of ``affine-invariant'' codes.
Affine-invariant codes are simply linear codes whose coordinates
are a vector space over a field and which are invariant under
affine-transformations of the coordinate space. Lifting takes codes
defined over a vector space of small dimension ... more >>>


TR12-149 | 8th November 2012
Alan Guo, Swastik Kopparty, Madhu Sudan

New affine-invariant codes from lifting

Comments: 1

In this work we explore error-correcting codes derived from
the ``lifting'' of ``affine-invariant'' codes.
Affine-invariant codes are simply linear codes whose coordinates
are a vector space over a field and which are invariant under
affine-transformations of the coordinate space. Lifting takes codes
defined over a vector space of small dimension ... more >>>


TR12-148 | 7th November 2012
Eli Ben-Sasson, Ariel Gabizon, Yohay Kaplan, Swastik Kopparty, Shubhangi Saraf

A new family of locally correctable codes based on degree-lifted algebraic geometry codes

Revisions: 1

We describe new constructions of error correcting codes, obtained by "degree-lifting" a short algebraic geometry (AG) base-code of block-length $q$ to a lifted-code of block-length $q^m$, for arbitrary integer $m$. The construction generalizes the way degree-$d$, univariate polynomials evaluated over the $q$-element field (also known as Reed-Solomon codes) are "lifted" ... more >>>


TR12-148 | 7th November 2012
Eli Ben-Sasson, Ariel Gabizon, Yohay Kaplan, Swastik Kopparty, Shubhangi Saraf

A new family of locally correctable codes based on degree-lifted algebraic geometry codes

Revisions: 1

We describe new constructions of error correcting codes, obtained by "degree-lifting" a short algebraic geometry (AG) base-code of block-length $q$ to a lifted-code of block-length $q^m$, for arbitrary integer $m$. The construction generalizes the way degree-$d$, univariate polynomials evaluated over the $q$-element field (also known as Reed-Solomon codes) are "lifted" ... more >>>


TR12-148 | 7th November 2012
Eli Ben-Sasson, Ariel Gabizon, Yohay Kaplan, Swastik Kopparty, Shubhangi Saraf

A new family of locally correctable codes based on degree-lifted algebraic geometry codes

Revisions: 1

We describe new constructions of error correcting codes, obtained by "degree-lifting" a short algebraic geometry (AG) base-code of block-length $q$ to a lifted-code of block-length $q^m$, for arbitrary integer $m$. The construction generalizes the way degree-$d$, univariate polynomials evaluated over the $q$-element field (also known as Reed-Solomon codes) are "lifted" ... more >>>


TR12-148 | 7th November 2012
Eli Ben-Sasson, Ariel Gabizon, Yohay Kaplan, Swastik Kopparty, Shubhangi Saraf

A new family of locally correctable codes based on degree-lifted algebraic geometry codes

Revisions: 1

We describe new constructions of error correcting codes, obtained by "degree-lifting" a short algebraic geometry (AG) base-code of block-length $q$ to a lifted-code of block-length $q^m$, for arbitrary integer $m$. The construction generalizes the way degree-$d$, univariate polynomials evaluated over the $q$-element field (also known as Reed-Solomon codes) are "lifted" ... more >>>


TR12-102 | 16th August 2012
Swastik Kopparty, Srikanth Srinivasan

Certifying Polynomials for $\mathrm{AC}^0[\oplus]$ circuits, with applications

In this paper, we introduce and develop the method of certifying polynomials for proving $\mathrm{AC}^0[\oplus]$ circuit lower bounds.

We use this method to show that Approximate Majority cannot be computed by $\mathrm{AC}^0[\oplus]$ circuits of size $n^{1+o(1)}$. This implies a separation between the power of $\mathrm{AC}^0[\oplus]$ circuits of near-linear size and ... more >>>


TR12-102 | 16th August 2012
Swastik Kopparty, Srikanth Srinivasan

Certifying Polynomials for $\mathrm{AC}^0[\oplus]$ circuits, with applications

In this paper, we introduce and develop the method of certifying polynomials for proving $\mathrm{AC}^0[\oplus]$ circuit lower bounds.

We use this method to show that Approximate Majority cannot be computed by $\mathrm{AC}^0[\oplus]$ circuits of size $n^{1+o(1)}$. This implies a separation between the power of $\mathrm{AC}^0[\oplus]$ circuits of near-linear size and ... more >>>


TR12-102 | 16th August 2012
Swastik Kopparty, Srikanth Srinivasan

Certifying Polynomials for $\mathrm{AC}^0[\oplus]$ circuits, with applications

In this paper, we introduce and develop the method of certifying polynomials for proving $\mathrm{AC}^0[\oplus]$ circuit lower bounds.

We use this method to show that Approximate Majority cannot be computed by $\mathrm{AC}^0[\oplus]$ circuits of size $n^{1+o(1)}$. This implies a separation between the power of $\mathrm{AC}^0[\oplus]$ circuits of near-linear size and ... 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 >>>


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 >>>


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 >>>


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 >>>


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 >>>


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 >>>


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 >>>


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 >>>


TR10-148 | 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 >>>


TR10-148 | 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 >>>


TR10-148 | 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 >>>


TR10-148 | 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 >>>


TR10-148 | 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 >>>


TR10-044 | 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 >>>


TR10-044 | 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 Hastad, 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 >>>


TR10-003 | 6th January 2010
Venkatesan Guruswami, Johan Hastad, 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 >>>


TR10-003 | 6th January 2010
Venkatesan Guruswami, Johan Hastad, 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 >>>


TR09-115 | 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 >>>


TR09-115 | 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 >>>


TR09-086 | 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 >>>


TR09-086 | 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 >>>


TR09-086 | 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 >>>


TR09-033 | 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 >>>


TR09-033 | 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 >>>


TR09-033 | 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 >>>


TR09-033 | 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.
\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 ... 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.
\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 ... 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.
\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 ... 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.
\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 ... 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 >>>


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 >>>


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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