We present new constructions of quasi-cyclic (QC) and generalized quasi-cyclic (GQC) codes from algebraic curves. Unlike previous approaches based on elliptic curves, our method applies to curves that are Kummer extensions of the rational function field, including hyperelliptic, norm-trace, and Hermitian curves. This allows QC codes with flexible co-index. Explicit parameter formulas are derived using known automorphism-group classifications.

翻译：本文提出了基于代数曲线构造准循环码与广义准循环码的新方法。与以往基于椭圆曲线的方案不同，本方法适用于有理函数域的Kummer扩张曲线，包括超椭圆曲线、范数迹曲线和Hermitian曲线。该框架支持构造具有灵活共指数的准循环码。通过利用已知的自同构群分类结果，我们推导出了显式的参数计算公式。