Characterization of non-special divisors of small degree on Kummer extensions and LCP codes

TL;DR

This paper characterizes non-special divisors of small degree on Kummer extensions and applies these findings to construct new algebraic geometry codes with linear complementary pairs.

Contribution

It provides an arithmetic criterion for identifying non-special divisors on Kummer extensions and explicitly determines such divisors in specific cases, leading to new LCP code families.

Findings

Arithmetic criterion for non-special divisors of degree g-1 and g

Explicit determination of non-special divisors in certain cases

Construction of new LCP algebraic geometry codes

Abstract

A recent construction of linear complementary pairs (LCPs) of algebraic geometry codes is intimately linked to the identification of non-special divisors of small degree within a function field over a finite field. Let $\mathbb{F}_q$ be the finite field of cardinality $q$. In this work, we consider a function field $F/\mathbb{F}_q$ of genus $g$ defined by a Kummer extension of type $y^m = f(x)$, where $f(x)$ is a polynomial in $\mathbb{F}_q[x]$. Based on the theory of generalized Weierstrass semigroups at several places, we provide an arithmetic criterion to identify all non-special divisors of degree $g-1$ and $g$ whose support is contained in a subset of the totally ramified places of the extension $F/\mathbb{F}_q(x)$. Furthermore, we explicitly determine all non-special divisors of degree $g-1$ in certain cases. Finally, we apply these results to provide explicit new families of LCPs algebraic geometry codes.