The automorphism groups of a maximal function field with the second largest genus and its algebraic geometry codes

TL;DR

This paper studies the automorphism groups of a special maximal function field with high genus and explores their impact on algebraic geometry codes, providing explicit examples and characterizations.

Contribution

It determines the automorphism groups of the Abdón--Torres maximal function field and related algebraic geometry codes, offering explicit equations and subgroup analyses.

Findings

Automorphism groups of the Abdón--Torres function field are characterized.

Automorphism groups of associated algebraic geometry codes are identified.

Explicit defining equations for a family of maximal function fields are provided.

Abstract

In this manuscript, we investigate the automorphism group of a maximal function field with the second largest possible genus over finite field of even characteristic, which is called the Abd\'on--Torres function field. As an application, we determine the automorphism groups of one-point algebraic geometry codes from such a maximal function field. It turns out that the automorphism groups of one-point algebraic geometry codes agree with that of the Abd\'on--Torres function field except for the trivial cases. Moreover, we provide a family of maximal function fields with explicit defining equations via considering fixed subfields with respect to some subgroups of automorphism group of the Abd\'on--Torres function field.