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