Explicit non-special divisors of small degree and LCD codes from Kummer extensions
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A linear code is linear complementary dual, or LCD, if and only if the intersection between the code and its dual is trivial. Introduced by Massey in 1992, LCD codes have attracted recent attention due to their application. In this vein, Mesnager, Tang, and Qi considered complementary dual algebraic geometric codes, giving several examples from low genus curves as well as instances using places of higher degree from Hermitian curves. In this paper, we consider the hull of an algebraic geometry code, meaning the intersection of the code and its dual. We demonstrate how LCD codes from Kummer extensions (and Hermitian curves in particular) may be defined using only rational points. Our primary tool is the explicit construction of non-special divisors of degrees g and g-1 on certain families of curves with many rational points. As a result, we provide algebraic geometric LCD codes from some maximal curves (such as the Hermitian code and a quotient) along with other curves (such as the norm-trace curve) supported by only rational points by appealing to Weierstrass semigroups.
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