On codes and matroids: minors, self-orthogonality, cycle nested and doubly-even matroids

Loucks, Weston Bret (2021) On codes and matroids: minors, self-orthogonality, cycle nested and doubly-even matroids. Masters thesis, Northern Arizona University.

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Abstract

The strong interplay between codes and matroids has generated a considerable interest in the discrete mathematics literature over several decades. While the main starting point in this interplay was the connection between the weight enumerator of a code and the Tutte polynomial of a matroid, many more connections have been established since. In this thesis, we explore these connections further, and introduce new notions for matroids using concepts from coding theory. Our main contributions can be summarized in two parts. In the first part, we consider the code-minor problem, which is something that has been inspired by matroid operations, but has not been properly investigated. We find necessary and sufficient conditions for a binary code to be a minor of another binary code, together with the corresponding results for matroids. We focus on special matrices for binary codes to get further results, using strings and substrings. In particular, we focus on establishing conditions for self-dual binary codes to have the extended binary Hamming or the extended binary Golay code as a minor. In the second part, we introduce the notions of cycle nested and doubly-even matroids using concepts of self-orthogonality from coding theory. In the binary case, we characterize the cocycle nested matroids and describe some properties of doubly-even matroids by relating them to doubly-even codes.

Item Type:	Thesis (Masters)
Publisher’s Statement:	© Copyright is held by the author. Digital access to this material is made possible by the Cline Library, Northern Arizona University. Further transmission, reproduction or presentation of protected items is prohibited except with permission of the author.
Keywords:	binary codes; code minors; Hamming code; matroids; self-dual; self-orthogonality
Subjects:	Q Science > QA Mathematics
NAU Depositing Author Academic Status:	Student
Department/Unit:	Graduate College > Theses and Dissertations College of the Environment, Forestry, and Natural Sciences > Mathematics and Statistics
Date Deposited:	03 Mar 2022 18:24
Last Modified:	03 Mar 2022 18:24
URI:	https://openknowledge.nau.edu/id/eprint/5788

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