Automorphisms of Hermitian Codes and Applications

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.