Archived
This content is available here strictly for research, reference, and/or recordkeeping and as such it may not be fully accessible. If you work or study at University of Kentucky and would like to request an accessible version, please use the SensusAccess Document Converter.
Author ORCID Identifier
Date Available
6-29-2021
Year of Publication
2021
Document Type
Doctoral Dissertation
Degree Name
Doctor of Philosophy (PhD)
College
Arts and Sciences
Department/School/Program
Mathematics
Faculty
Dr. Heide Gluesing-Luerssen
Faculty
Dr. Benjamin Braun
Abstract
Cyclic orbit codes are subspace codes generated by the action of the Singer subgroup Fqn* on an Fq-subspace U of Fqn. The weight distribution of a code is the vector whose ith entry is the number of codewords with distance i to a fixed reference space in the code. My dissertation investigates the structure of the weight distribution for cyclic orbit codes. We show that for full-length orbit codes with maximal possible distance the weight distribution depends only on q,n and the dimension of U. For full-length orbit codes with lower minimum distance, we provide partial results towards a characterization of the weight distribution, especially in the case that any two codewords intersect in a space of dimension at most 2. We also briefly address the weight distribution of a union of full-length orbit codes with maximum distance.
A related problem is to find the automorphism group of a cyclic orbit code, which plays a role in determining the isometry classes of the set of all cyclic orbit codes. First we show that the automorphism group of a cyclic orbit code is contained in the normalizer of the Singer subgroup if the orbit is generated by a subspace that is not contained in a proper subfield of Fqn. We then generalize to orbits under the normalizer of the Singer subgroup, although in this setup there is a remaining exceptional case. Finally, we can characterize linear isometries between such codes.
Digital Object Identifier (DOI)
https://doi.org/10.13023/etd.2021.246
Recommended Citation
Lehmann, Hunter, "Weight Distributions, Automorphisms, and Isometries of Cyclic Orbit Codes" (2021). Theses and Dissertations--Mathematics. 84.
https://uknowledge.uky.edu/math_etds/84
