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

TL;DR

The paper addresses the construction and characterization of algebraic geometry codes whose hulls can be explicitly described using only rational places, focusing on LCD codes and hull control. It develops explicit non-special divisors of small degree on Kummer extensions via Weierstrass semigroups, enabling hulls C(D,G) with hulls expressible as C(D, gcd(G,H)) under canonical-type relations. By leveraging maximal function fields (notably Hermitian) and the gcd/lmd framework, the authors obtain LCD codes through divisors of degree $g-1$ and provide concrete examples with rational places only. This work extends prior efforts by delivering constructive hull descriptions and LCD constructions without resorting to monomial equivalences, with implications for cryptography and coding theory using maximal and near-maximal function fields.

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.

Figures (1)

• Figure 1: Impact of $\deg A$ on the index of speciality of a divisor $A$ on a function field $F$ of genus $g$

Theorems & Definitions (29)

• Lemma 1
• Theorem 2
• proof
• Corollary 3
• Lemma 4
• Proposition 5
• proof
• Proposition 6: oneandtwo, kummer
• Lemma 7
• proof