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Probabilistic and polynomial methods in additive combinatorics and coding theory

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Title
Probabilistic and polynomial methods in additive combinatorics and coding theory
Name (type = personal)
NamePart (type = family)
Kim
NamePart (type = given)
John
NamePart (type = date)
1988-
DisplayForm
John Kim
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author
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Kopparty
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Swastik
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Swastik Kopparty
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Advisory Committee
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chair
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Saks
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Michael
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Michael Saks
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Advisory Committee
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internal member
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Zeilberger
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Doron
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Doron Zeilberger
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Advisory Committee
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internal member
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Ben-Sasson
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Eli
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Eli Ben-Sasson
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Advisory Committee
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outside member
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Rutgers University
Role
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degree grantor
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Graduate School - New Brunswick
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school
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theses
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2016
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2016-05
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2016
Place
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xx
Language
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eng
Abstract (type = abstract)
We present various applications of the probabilistic method and polynomial method in additive combinatorics and coding theory. We first study the effect of addition on the Hamming weight of a positive integer. Consider the first 2n positive integers, and fix an alpha among them. We show that if the binary representation of alpha consists of Theta(n) blocks of zeros and ones, then addition by alpha causes a constant fraction of low Hamming weight integers to become high Hamming weight integers. Our result implies that powering by alpha composed of many blocks requires exponential-size, bounded-depth arithmetic circuits over F_2. We also prove a version of the Cauchy-Davenport theorem for general linear maps. For subsets A, B of the finite field F_p, the classical Cauchy-Davenport theorem gives a lower bound for the size of the sumset A + B in terms of the sizes of the sets A and B. Our theorem considers a general linear map L: F_p^n to F_p^n, and subsets A_1, ..., A_n of F_p, and gives a lower bound on the size of L(A_1 X A_2 X ... X A_n) in terms of the sizes of the sets A_1, ..., A_n. Our proof uses Alon's Combinatorial Nullstellensatz and a variation of the polynomial method. Lastly, 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 achieve this only if the set S has some very special algebraic structure, or if the degree d is significantly smaller than |S|. We also give a near-linear time algorithm, which is based on tools from list-decoding, to decode these codes from nearly half their minimum distance, provided d < (1 - epsilon)|S| for constant epsilon > 0.
Subject (authority = RUETD)
Topic
Mathematics
Subject (authority = ETD-LCSH)
Topic
Combinatorial probabilities
RelatedItem (type = host)
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Title
Rutgers University Electronic Theses and Dissertations
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ETD
Identifier
ETD_7272
PhysicalDescription
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electronic resource
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application/pdf
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text/xml
Extent
1 online resource (viii, 88 p. : ill.)
Note (type = degree)
Ph.D.
Note (type = bibliography)
Includes bibliographical references
Note (type = statement of responsibility)
by John Kim
RelatedItem (type = host)
TitleInfo
Title
Graduate School - New Brunswick Electronic Theses and Dissertations
Identifier (type = local)
rucore19991600001
Location
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NjNbRU
Identifier (type = doi)
doi:10.7282/T3CF9S8H
Genre (authority = ExL-Esploro)
ETD doctoral
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The author owns the copyright to this work.
RightsHolder (type = personal)
Name
FamilyName
Kim
GivenName
John
Role
Copyright Holder
RightsEvent
Type
Permission or license
DateTime (encoding = w3cdtf); (qualifier = exact); (point = start)
2016-04-15 13:56:14
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Name
John Kim
Role
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Affiliation
Rutgers University. Graduate School - New Brunswick
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Author Agreement License
Detail
I hereby grant to the Rutgers University Libraries and to my school the non-exclusive right to archive, reproduce and distribute my thesis or dissertation, in whole or in part, and/or my abstract, in whole or in part, in and from an electronic format, subject to the release date subsequently stipulated in this submittal form and approved by my school. I represent and stipulate that the thesis or dissertation and its abstract are my original work, that they do not infringe or violate any rights of others, and that I make these grants as the sole owner of the rights to my thesis or dissertation and its abstract. I represent that I have obtained written permissions, when necessary, from the owner(s) of each third party copyrighted matter to be included in my thesis or dissertation and will supply copies of such upon request by my school. I acknowledge that RU ETD and my school will not distribute my thesis or dissertation or its abstract if, in their reasonable judgment, they believe all such rights have not been secured. I acknowledge that I retain ownership rights to the copyright of my work. I also retain the right to use all or part of this thesis or dissertation in future works, such as articles or books.
Copyright
Status
Copyright protected
Availability
Status
Open
Reason
Permission or license
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2016-04-15T13:36:03
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