Graduation Year
2024
Document Type
Dissertation
Degree
Ph.D.
Degree Name
Doctor of Philosophy (Ph.D.)
Degree Granting Department
Mathematics and Statistics
Major Professor
Giacomo Micheli, Ph.D.
Co-Major Professor
Lukas Kölsch, Ph.D.
Committee Member
Jean-François Biasse, Ph.D.
Committee Member
Xiang-dong Hou, Ph.D.
Committee Member
Joachim Rosenthal, Ph.D.
Keywords
chebotarev density theorem, good polynomials, hierarchical locally recoverable codes, locally recoverable codes
Abstract
To keep up with the ever-growing demand for reliable and efficient availability of data,locally recoverable codes (LRCs) have been the focus of much study due to their applications in cloud and distributed storage systems. A fundamental construction of LRCs was given in [36] based on polynomials which relied on the existence of r-good polynomials. In the same paper some constructions of good polynomials were given, but these constructions did not cover every configuration of parameters. Naturally this led to research into constructing good polynomials for what was not addressed in [36], but new ponderings were also posed, such as the following: for a locally recoverable code with fixed parameters, are some good polynomials better than others? In [29], Micheli approached this topic using machinery from Galois theory and defined more generally (r, ℓ)-good polynomials and showed that maximizing ℓ (for a given locality r) yields a code with higher dimension. In this dissertation, we determine the optimum value of ℓ for any (r, ℓ)-good polynomials of degree up to 5. We then provide an explicit construction of a newer class of LRCs called hierarchical locally recoverable codes (HLRCs) and prove that our construction yields codes with higher dimension than existing literature in some regimes of parameters. This dissertation is intended to be self-contained. As a result, the first few chapters address the fundamental background in coding theory, algebraic number theory, etc., needed to understand the aforementioned results.
Scholar Commons Citation
Dukes, Austin, "Optimal Selection of Good Polynomials and Constructions of Locally Recoverable Codes via Galois Theory" (2024). USF Tampa Graduate Theses and Dissertations.
https://digitalcommons.usf.edu/etd/10503
