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.

翻译：本文考虑代数几何码的壳，即码与其对偶码的交集。我们展示了如何仅利用库默尔扩张（特别是埃尔米特函数域）的有理点来定义壳为代数几何码的码。核心工具是在具有众多有理点的特定函数域族上显式构造次数为$g$和$g-1$的非特殊因子，这通过韦尔斯特拉斯半群实现。我们给出了壳具有指定维数的显式代数几何码，进而仅利用有理点从埃尔米特函数域（及其他域）构造出线性互补对偶代数几何码，并解决了Ballet与Le Brigand针对特定函数域提出的开放问题。这些结果补充了Mesnager、Tang和Qi利用低亏格函数域以及高阶埃尔米特函数域点构造线性互补对偶码的早期工作，也完善了Carlet、Mesnager、Tang、Qi与Pellikaan通过单项等价性之外的方法直接提供具有LCD性质的显式代数几何码的研究。