The automorphism groups of a maximal function field with the second largest genus and its algebraic geometry codes
Xubin Hu, Liming Ma
TL;DR
The paper addresses the problem of determining the automorphism group of the Abdón--Torres maximal curve $Y_2$ over ${\mathbb F}_{q^2}$, which has the second largest genus among maximal curves, and its implications for one-point algebraic geometry codes. It develops a detailed analysis of Weierstrass semigroups at all rational places, identifies the stabilizer of the infinite place, and then establishes the full automorphism group as a semidirect product $C \ltimes N$ with explicit parameterizations of automorphisms. The authors further construct explicit fixed subfields $Y_2^G$ under abelian subgroups, giving defining equations and genus data for these Galois subfields, and apply these results to completely determine the automorphism groups of one-point AG codes $C_m$ from $Y_2$ in several regimes of $m$, including cases equal to the full curve automorphism group, affine-extended groups, and symmetric groups. The work advances both the structural understanding of maximal curves and the symmetry properties of associated algebraic geometry codes, with concrete equations for fixed fields and clear mappings between curve automorphisms and code automorphisms.
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 Abdón--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 Abdón--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 Abdón--Torres function field.