Pure Gaps at Many Places and Multi-point AG Codes from Arbitrary Kummer Extensions

TL;DR

This paper develops a method to identify pure gaps at many places in Kummer extensions and uses these to construct algebraic geometry codes with improved parameters, including a record-setting code.

Contribution

It introduces a simple, efficient approach to find pure gaps at multiple places in Kummer extensions and fully characterizes these gaps in specific cases, enabling better code construction.

Findings

Method to find all pure gaps at many places in Kummer extensions

Explicit description of pure gaps when parameters are equal

Construction of a new record algebraic geometry code

Abstract

For a Kummer extension defined by the affine equation $y^{m}=\prod_{i=1}^{r} (x-\a_i)^{\lambda_i}$ over an algebraic extension $K$ of a finite field $\fq$, where $\la_i\in \Z\backslash\{0\}$ for $1\leq i\leq r$, $\gcd(m,q) = 1$, and $\a_1,\cdots,\a_r\in K$ are pairwise distinct elements, we propose a simple and efficient method to find all pure gaps at many totally ramified places. We introduce a bottom set of pure gaps and indicate that the set of pure gaps is completely determined by the bottom set. Furthermore, we demonstrate that a pure gap can be deduced from a known pure gap by easily verifying only one inequality. Then, in the case where $\lambda_1 = \lambda_2 = \cdots = \lambda_r$, we fully determine an explicit description of the set of pure gaps at many totally ramified places, This includes the scenario in which the set of these places contains the infinite place. Finally, we apply these results to construct multi-point algebraic geometry codes with good parameters. As one of the examples, a presented code with parameters $[74, 60, \geq 10]$ over $\mathbb{F}_{25}$ yields a new record.