On Automorphisms and Subtowers of an asymptotically optimal Tower of Function Fields

TL;DR

This paper studies the automorphism groups of an asymptotically optimal tower of function fields, revealing structural properties and a new subtower that improves code construction beyond the Gilbert-Varshamov bound.

Contribution

It provides a detailed description of the automorphism group's decomposition of the Garcia-Stichtenoth tower and introduces a new asymptotically optimal non-Galois subtower.

Findings

Decomposition group acts on algebraic-geometric codes exceeding the Gilbert-Varshamov bound.

Fixed fields form an asymptotically optimal non-Galois subtower.

New results on rational places and Weierstra{ extss}ss semigroups of the subtower.

Abstract

In this article we investigate the automorphism group of an asymptotically optimal tower of function fields introduced by Garcia and Stichtenoth. In particular we provide a detailed description of the decomposition group of some rational places. This group acts on the algebraic-geometric standard codes obtained by the Garcia-Stichtenoth tower exceeding the Gilbert-Varshamov bound. The fields fixed by the decomposition groups form an asymptotically optimal non-Galois subtower, which has been first found by Bezerra and Garcia and yields an improvement for computing codes above the Gilbert-Varshamov bound. In this article we also describe its proportionality to the Garcia-Stichtenoth tower and obtain new precise results on its rational places and their Weierstra{\ss} semigroups.