In recent years, linear complementary pairs (LCP) of codes and linear complementary dual (LCD) codes have gained significant attention due to their applications in coding theory and cryptography. In this work, we construct explicit LCPs of codes and LCD codes from function fields of genus $g \geq 1$. To accomplish this, we present pairs of suitable divisors giving rise to non-special divisors of degree $g-1$ in the function field. The results are applied in constructing LCPs of algebraic geometry codes and LCD algebraic geometry (AG) codes in Kummer extensions, hyperelliptic function fields, and elliptic curves.

翻译：近年来，线性互补对（LCP）码与线性互补对偶（LCD）码因其在编码理论与密码学中的应用而受到广泛关注。本文从亏格 $g \geq 1$ 的函数域出发，显式构造了码的线性互补对与线性互补对偶码。为实现这一目标，我们提出了若干对合适的除子，这些除子在函数域中生成次数为 $g-1$ 的非特殊除子。所得结果被应用于在Kummer扩张、超椭圆函数域及椭圆曲线上构造代数几何码的线性互补对以及LCD代数几何（AG）码。