Automorphisms of Hermitian Codes and Applications
| dc.contributor.author | Lichtenwalner, Monica M. | en |
| dc.contributor.committeechair | Matthews, Gretchen L. | en |
| dc.contributor.committeecochair | López, Hiram H. | en |
| dc.contributor.committeemember | Cotardo, Giuseppe | en |
| dc.contributor.department | Mathematics | en |
| dc.date.accessioned | 2026-06-17T16:26:24Z | en |
| dc.date.available | 2026-06-17T16:26:24Z | en |
| dc.date.issued | 2026-05-07 | en |
| dc.description.abstract | Code automorphisms have long been studied due to their many applications. Decoding, erasure recovery, local decodability, and local correctability can all benefit from codes with rich automorphism groups. Rigid codes, or codes that have a trivial automorphism group, are also interesting, mainly for applications to code-based post-quantum cryptography. It has been shown that, with high probability, a random linear code has a trivial automorphism group, but it can be difficult to explicitly construct a code that is rigid. Automorphisms of one-point Hermitian codes were first studied by Xing in 1995. In this thesis, we determine the automorphisms of multi-point Hermitian codes that are induced by certain curve automorphisms. We also introduce the notion of curve-rigid codes, and we present a family of three-point Hermitian codes that are curve-rigid. Lastly, we demonstrate how we can use the automorphism group of a one-point Hermitian or norm-trace code to correct certain error patterns via permutation decoding, which is a decoding procedure developed by Prange in 1962 and extended by MacWilliams in 1964. | en |
| dc.description.abstractgeneral | Coding theory addresses the setting in which one party has information that they wish to transmit over some channel to a second party. When sending information over the channel, there is the possibility that errors may occur, and the second party may receive incorrect information. To counteract this, we add some form of structured redundancy to the information we wish to transmit, so that the receiving party can detect, and possibly correct, errors that may have occurred during transmission. The set of messages with the added redundancy is referred to as a code. The set of symmetries, or automorphisms, of codes is studied for a variety of reasons, as they have many practical applications, including decoding and erasure recovery. Codes that lack symmetry are also interesting, mainly for applications to post-quantum cryptography. As shown by Lefmann, Phelps, and Rodl in 1993, a random linear code, with high probability, has no automorphisms, but it can be difficult to explicitly construct codes with no automorphisms. In this thesis, we introduce a family of Hermitian codes that are curve-rigid, meaning they have no automorphisms induced by the underlying curve. We also demonstrate how the symmetries of Hermitian and norm-trace codes can be used during the decoding process to correct certain error patterns. | en |
| dc.description.degree | Master of Science | en |
| dc.format.medium | ETD | en |
| dc.format.mimetype | application/pdf | en |
| dc.identifier.uri | https://hdl.handle.net/10919/143447 | en |
| dc.publisher | Virginia Tech | en |
| dc.rights | In Copyright | en |
| dc.rights.uri | http://rightsstatements.org/vocab/InC/1.0/ | en |
| dc.subject | Hermitian codes | en |
| dc.subject | code automorphisms | en |
| dc.subject | permutation decoding | en |
| dc.subject | algebraic geometry codes | en |
| dc.subject | Hermitian curve | en |
| dc.subject | norm-trace curve | en |
| dc.title | Automorphisms of Hermitian Codes and Applications | en |
| dc.type | Thesis | en |
| dc.type.dcmitype | Text | en |
| thesis.degree.discipline | Mathematics | en |
| thesis.degree.grantor | Virginia Polytechnic Institute and State University | en |
| thesis.degree.level | masters | en |
| thesis.degree.name | Master of Science | en |