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Finite Groups, Polymatroids, and Error-Correcting Codes
Finite Groups, Polymatroids, and Error-Correcting Codes
상세정보
- 자료유형
- 학위논문 서양
- 최종처리일시
- 20250211151324
- ISBN
- 9798382840345
- DDC
- 510
- 서명/저자
- Finite Groups, Polymatroids, and Error-Correcting Codes
- 발행사항
- [Sl] : Cornell University, 2024
- 발행사항
- Ann Arbor : ProQuest Dissertations & Theses, 2024
- 형태사항
- 71 p
- 주기사항
- Source: Dissertations Abstracts International, Volume: 85-12, Section: B.
- 주기사항
- Advisor: Swartz, Edward.
- 학위논문주기
- Thesis (Ph.D.)--Cornell University, 2024.
- 초록/해제
- 요약In 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.
- 일반주제명
- Mathematics
- 일반주제명
- Applied mathematics
- 키워드
- Finite groups
- 키워드
- Matroids
- 키워드
- Polymatroids
- 기타저자
- Cornell University Mathematics
- 기본자료저록
- Dissertations Abstracts International. 85-12B.
- 전자적 위치 및 접속
- 로그인 후 원문을 볼 수 있습니다.
MARC
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■020 ▼a9798382840345
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■040 ▼aMiAaPQ▼cMiAaPQ
■0820 ▼a510
■1001 ▼aWentworth-Nice, Prairie Elizabeth.▼0(orcid)0000-0002-0106-3738
■24510▼aFinite Groups, Polymatroids, and Error-Correcting Codes
■260 ▼a[Sl]▼bCornell University▼c2024
■260 1▼aAnn Arbor▼bProQuest Dissertations & Theses▼c2024
■300 ▼a71 p
■500 ▼aSource: Dissertations Abstracts International, Volume: 85-12, Section: B.
■500 ▼aAdvisor: Swartz, Edward.
■5021 ▼aThesis (Ph.D.)--Cornell University, 2024.
■520 ▼aIn 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.
■590 ▼aSchool code: 0058.
■650 4▼aMathematics
■650 4▼aApplied mathematics
■653 ▼aError-correcting codes
■653 ▼aFinite groups
■653 ▼aMatroids
■653 ▼aPolymatroids
■690 ▼a0405
■690 ▼a0364
■71020▼aCornell University▼bMathematics.
■7730 ▼tDissertations Abstracts International▼g85-12B.
■790 ▼a0058
■791 ▼aPh.D.
■792 ▼a2024
■793 ▼aEnglish
■85640▼uhttp://www.riss.kr/pdu/ddodLink.do?id=T17161201▼nKERIS▼z이 자료의 원문은 한국교육학술정보원에서 제공합니다.
