The Order Bound on the Minimum Distance of the One-Point Codes Associated to a Garcia-Stichtenoth Tower of Function Fields

TL;DR

This paper provides an explicit formula for the order bound on the minimum distance of one-point codes derived from Garcia-Stichtenoth towers, enhancing understanding of their error-correcting capabilities.

Contribution

It introduces a non-recursive description of Weierstrass semigroups for these towers, enabling explicit calculation of the Feng-Rao bound on code minimum distance.

Findings

Derived explicit formula for the order bound

Provided non-recursive semigroup descriptions

Enhanced minimum distance estimation for codes

Abstract

Garcia and Stichtenoth discovered two towers of function fields that meet the Drinfeld-Vl\u{a}du\c{t} bound on the ratio of the number of points to the genus. For one of these towers, Garcia, Pellikaan and Torres derived a recursive description of the Weierstrass semigroups associated to a tower of points on the associated curves. In this article, a non-recursive description of the semigroups is given and from this the enumeration of each of the semigroups is derived as well as its inverse. This enables us to find an explicit formula for the order (Feng-Rao) bound on the minimum distance of the associated one-point codes.