Codes from curves over finite fields were first developed in the late 1970's by V. D. Goppa and are known as algebraic geometry codes. Since that time, the construction has been tailored to fit particular applications, such as erasure recovery and error correction using less received information than in the classical case. The Hermitian-lifted code construction of L'opez, Malmskog, Matthews, Piñero-González, and Wootters (2021) provides codes from the Hermitian curve over $F_{q^2}$ which have the same locality as the well-known one-point Hermitian codes but with a rate bounded below by a positive constant independent of the field size. However, obtaining explicit expressions for the code is challenging.
In this dissertation, we consider codes from norm-trace curves, which are a generalization of the Hermitian curve. We develop norm-trace-lifted codes and demonstrate an explicit basis of the codes. We then consider fractional decoding of codes from norm-trace curves, extending the results obtained for codes from the Hermitian curve by Matthews, Murphy, and Santos (2021). / Doctor of Philosophy / Coding theory focuses on recovering information, whether that data is corrupted and changed (called an error) or is simply lost (called an erasure). Classical codes achieve this goal by accessing all received symbols. Because long codes, meaning those with many symbols, are common in applications, it is useful for codes to be able to correct errors and recover erasures by accessing less information than classical codes allow. That is the focus of this dissertation.
Codes with locality are designed for erasure recovery using fewer symbols than in the classical case. Such codes are said to have locality $r$ and availability $s$ if each symbol can be recovered from $s$ disjoint sets of $r$ other symbols. Algebraic curves, such as the Hermitian curve or the more general norm-trace curves, offer a natural structure for designing codes with locality. This is done by considering lines intersected with the curve to form repair groups, which are sets of $r+1$ points where the information from one point can be recovered using the rest of the points in the repair group.
An error correction method which uses less data than the classical case is that of fractional decoding. Fractional decoding takes advantage of algebraic properties of the field trace to correct errors by downloading only a $lambda$-proportion of the received information, where $lambda < 1$.
In this work, we consider a new family of codes resulting from norm-trace curves, and study their locality and availability, as well as apply the ideas of fractional decoding to these codes.
| Identifer | oai:union.ndltd.org:VTETD/oai:vtechworks.lib.vt.edu:10919/109538 |
| Date | 04 April 2022 |
| Creators | Murphy, Aidan W. |
| Contributors | Mathematics, Matthews, Gretchen L., Manganiello, Felice, Morrison, Travis William, Mihalcea, Constantin Leonardo |
| Publisher | Virginia Tech |
| Source Sets | Virginia Tech Theses and Dissertation |
| Language | English |
| Detected Language | English |
| Type | Dissertation |
| Format | ETD, application/pdf |
| Rights | In Copyright, http://rightsstatements.org/vocab/InC/1.0/ |
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