Recently, constructions of linear complementary pairs (LCPs) of codes and linear complementary dual (LCD) codes on function fields have attracted considerable attention due to the wide range of applications of these codes. Such constructions rely on non-special divisors of degrees $g$ and $g-1$. In this work, we investigate Kummer extensions defined by $y^m = f(x)$ with $f(x)\in\mathbb{F}_q(x)$ and establish an arithmetic characterization of non-special divisors whose support can contain non-totally ramified places. Based on this characterization, we explicitly construct non-special divisors of degree $g-1$ on the GK curve. Moreover, utilizing pure gaps, we explicitly provide several families of effective non-special divisors of degree $g$ on Kummer extensions with the same multiplicities. We then develop a general framework for constructing LCPs of algebraic geometry (AG) codes on Kummer extensions. By virtue of canonical divisors, we show that the security parameters of LCPs of AG codes can be determined within this framework, which also enables the construction of LCD AG codes. Finally, we illustrate our results with representative examples, including LCPs of codes on the GK curve and LCD codes on quotients of the Hermitian curve.

翻译：最近，由于线性互补对（LCPs）和线性互补对偶（LCD）码在函数域上的构造因其广泛的应用而备受关注。此类构造依赖于次数为$g$和$g-1$的非特殊除子。本文中，我们研究由$y^m = f(x)$（其中$f(x)\in\mathbb{F}_q(x)$）定义的Kummer扩张，并建立一种算术特征，用以刻画支撑集可包含非完全分歧点的非特殊除子。基于该特征，我们在GK曲线上明确构造了次数为$g-1$的非特殊除子。此外，利用纯间隙，我们显式给出了Kummer扩张上若干族具有相同重数的有效次数为$g$的非特殊除子。随后，我们发展了一个一般性框架，用于在Kummer扩张上构造代数几何（AG）码的LCPs。凭借典范除子，我们证明在此框架内可以确定AG码LCPs的安全参数，这同时也使得LCD AG码的构造成为可能。最后，我们通过代表性实例（包括GK曲线上的码的LCPs和Hermitian曲线商上的LCD码）阐明我们的结果。