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：本文研究代数几何码的壳，即码与其对偶的交集。我们证明如何利用库默尔扩张（特别是埃尔米特函数域）的仅有有理点来定义壳为代数几何码的码。我们的主要工具是通过魏尔斯特拉斯半群，在具有多个有理点的特定函数域族上显式构造次数为$g$和$g-1$的非特殊除子。我们提供具有指定维数壳的显式代数几何码，在此过程中仅使用有理点从埃尔米特函数域（及其他函数域）构造线性互补对偶代数几何码，并解答了Ballet和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使用埃尔米特函数域更高度数点构造线性互补对偶(LCD)码的工作，也补充了Carlet、Mesnager、Tang、Qi和Pellikaan通过单式等价之外的方法提供具有LCD性质的显式代数几何码的研究。