Date Available
1-28-2019
Year of Publication
2019
Degree Name
Doctor of Philosophy (PhD)
Document Type
Doctoral Dissertation
College
Arts and Sciences
Department/School/Program
Mathematics
First Advisor
Dr. Heide Gluesing-Luerssen
Abstract
In classical and quantum information theory there are different types of error-correcting codes being used. We study the equivalence of codes via a classification of their isometries. The isometries of various codes over Frobenius alphabets endowed with various weights typically have a rich and predictable structure. On the other hand, when the alphabet is not Frobenius the isometry group behaves unpredictably. We use character theory to develop a duality theory of partitions over Frobenius bimodules, which is then used to study the equivalence of codes. We also consider instances of codes over non-Frobenius alphabets and establish their isometry groups. Secondly, we focus on quantum stabilizer codes over local Frobenius rings. We estimate their minimum distance and conjecture that they do not underperform quantum stabilizer codes over fields. We introduce symplectic isometries. Isometry groups of binary quantum stabilizer codes are established and then applied to the LU-LC conjecture.
Digital Object Identifier (DOI)
https://doi.org/10.13023/etd.2019.014
Recommended Citation
Pllaha, Tefjol, "Equivalence of Classical and Quantum Codes" (2019). Theses and Dissertations--Mathematics. 59.
https://uknowledge.uky.edu/math_etds/59
