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Finite Groups, Polymatroids, and Error-Correcting Codes /

Record Type:	Electronic resources : Monograph/item
Title/Author:	Finite Groups, Polymatroids, and Error-Correcting Codes /Prairie Elizabeth Wentworth-Nice.
Author:	Wentworth-Nice, Prairie Elizabeth,
Description:	1 electronic resource (71 pages)
Notes:	Source: Dissertations Abstracts International, Volume: 85-12, Section: B.
Notes:	Advisors: Swartz, Edward Committee members: Meszaros, Karola; Aguiar, Marcelo.
Contained By:	Dissertations Abstracts International85-12B.
Subject:	Mathematics.
Online resource:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=31239758
ISBN:	9798382840345

Finite Groups, Polymatroids, and Error-Correcting Codes /
Wentworth-Nice, Prairie Elizabeth,

Finite Groups, Polymatroids, and Error-Correcting Codes /Prairie Elizabeth Wentworth-Nice. - 1 electronic resource (71 pages)

Source: Dissertations Abstracts International, Volume: 85-12, Section: B.

In 1962, Jesse MacWilliams published formulas for linear codes that, among other applications, were incredibly valuable in the study of self-dual codes. Now called the MacWilliams Identities, her results relate the weight and complete weight enumerators of a code to those of its dual code. Similar identities have been proven to exist for many other types of codes. In 2013, Dougherty, Sole, and Kim published a list of fundamental open questions in coding theory. Among them, Open Question 4.3: "Is there a duality and MacWilliams formula for codes over non-Abelian groups?" In the latter half of this dissertation, we propose a duality for nonabelian group codes in terms of the irreducible representations of the group. We show that there is a Greene's Theorem and MacWilliams Identities which hold for this duality.This notion of a dual for nonabelian groups stems from a recent generalization of the theory of matroids representable over finite fields to finite groups and polymatroids. In the first half of this dissertation we describe this generalization and, given a finite group Γ, begin the characterization of polymatroids representable over Γ. We show that there is a unique excluded minor for matroids representable over nonabelian groups. In addition, we make progress towards describing which matroids are representable over abelian groups, and give some representability conditions for polymatroids over groups isomorphic to direct products.

English

ISBN: 9798382840345Subjects--Topical Terms:

184409
Mathematics.
Subjects--Index Terms:

Error-correcting codes

Finite Groups, Polymatroids, and Error-Correcting Codes /
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520 $a In 1962, Jesse MacWilliams published formulas for linear codes that, among other applications, were incredibly valuable in the study of self-dual codes. Now called the MacWilliams Identities, her results relate the weight and complete weight enumerators of a code to those of its dual code. Similar identities have been proven to exist for many other types of codes. In 2013, Dougherty, Sole, and Kim published a list of fundamental open questions in coding theory. Among them, Open Question 4.3: "Is there a duality and MacWilliams formula for codes over non-Abelian groups?" In the latter half of this dissertation, we propose a duality for nonabelian group codes in terms of the irreducible representations of the group. We show that there is a Greene's Theorem and MacWilliams Identities which hold for this duality.This notion of a dual for nonabelian groups stems from a recent generalization of the theory of matroids representable over finite fields to finite groups and polymatroids. In the first half of this dissertation we describe this generalization and, given a finite group Γ, begin the characterization of polymatroids representable over Γ. We show that there is a unique excluded minor for matroids representable over nonabelian groups. In addition, we make progress towards describing which matroids are representable over abelian groups, and give some representability conditions for polymatroids over groups isomorphic to direct products.
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650 4 $a Mathematics. $3 184409
650 4 $a Applied mathematics. $3 377601
653 $a Error-correcting codes
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653 $a Matroids
653 $a Polymatroids
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690 $a 0364
710 2 $a Cornell University. $b Mathematics. $e degree granting institution. $3 990072
720 1 $a Swartz, Edward $e degree supervisor.
773 0 $t Dissertations Abstracts International $g 85-12B.
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791 $a Ph.D.
792 $a 2024
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