Explicit non-special divisors of small degree, algebraic geometric hulls, and LCD codes from Kummer extensions

TL;DR

This paper constructs algebraic geometry codes with specified hull dimensions using explicit non-special divisors on Kummer extensions, advancing the understanding of LCD codes from algebraic curves with many rational places.

Contribution

It introduces a method to define codes with hulls of desired dimensions using only rational places of Kummer extensions, including solutions to an open problem for specific function fields.

Findings

Constructed explicit algebraic geometry codes with prescribed hull dimensions.

Produced linearly complementary dual (LCD) codes from Hermitian function fields.

Provided an explicit answer to an open question by Ballet and Le Brigand.

Abstract

In this paper, we consider the hull of an algebraic geometry code, meaning the intersection of the code and its dual. We demonstrate how codes whose hulls are algebraic geometry codes may be defined using only rational places of Kummer extensions (and Hermitian function fields in particular). Our primary tool is explicitly constructing non-special divisors of degrees $g$ and $g-1$ on certain families of function fields with many rational places, accomplished by appealing to Weierstrass semigroups. We provide explicit algebraic geometry codes with hulls of specified dimensions, producing along the way linearly complementary dual algebraic geometric codes from the Hermitian function field (among others) using only rational places and an answer to an open question posed by Ballet and Le Brigand for particular function fields. These results complement earlier work by Mesnager, Tang, and Qi that use lower-genus function fields as well as instances using places of a higher degree from Hermitian function fields to construct linearly complementary dual (LCD) codes and that of Carlet, Mesnager, Tang, Qi, and Pellikaan to provide explicit algebraic geometry codes with the LCD property rather than obtaining codes via monomial equivalences.