Explicit Non-Special Divisors of Small Degree and LCD Codes from Kummer Extensions.

In this paper, we consider the hull of an algebraic geometry code, meaningthe intersection of the code and its dual. We demonstrate how codes whose hullsare algebraic geometry codes may be defined using only rational places ofKummer extensions (and Hermitian function fields in particular). Our primarytool is explicitly constructing non-special divisors of degrees g and g-1on certain families of function fields with many rational places, accomplishedby appealing to Weierstrass semigroups. We provide explicit algebraic geometrycodes with hulls of specified dimensions, producing along the way linearlycomplementary dual algebraic geometric codes from the Hermitian function field(among others) using only rational places and an answer to an open questionposed by Ballet and Le Brigand for particular function fields. These resultscomplement earlier work by Mesnager, Tang, and Qi that use lower-genus functionfields as well as instances using places of a higher degree from Hermitianfunction fields to construct linearly complementary dual (LCD) codes and thatof Carlet, Mesnager, Tang, Qi, and Pellikaan to provide explicit algebraicgeometry codes with the LCD property rather than obtaining codes via monomialequivalences.