JOONGBU UNIVERSITY LIBRARY

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Finite Groups, Polymatroids, and Error-Correcting Codes.

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

Abstracts/Etc: 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.

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