Codes from norm-trace curves: local recovery and fractional decoding

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_q² 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).

Name thesis:degree_name

Doctor of Philosophy

Level thesis:degree_level

doctoral

Discipline thesis:degree_discipline

Mathematics

Department dc:contributor.department

Mathematics

Grantor dc:publisher

Virginia Tech

Year dc:date.issued

2022

Author dc:creator

• Murphy, Aidan W.

Chair dc:contributor.committeechair

• Matthews, Gretchen L.

Committee members dc:contributor.committeemember

• Manganiello, Felice
• Morrison, Travis William
• Mihalcea, Constantin Leonardo

• algebraic geometry code
• locally recoverable code
• fractional decoding
• norm-trace curve

Statement dc:rights

• In Copyright

Licence dc:rights.uri

• http://rightsstatements.org/vocab/InC/1.0/

Language dc:language.iso

en

Dc Identifier Other

vt_gsexam:34234

OAI identifier oai:identifier

oai:vtechworks.lib.vt.edu:10919/109538