  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.
