Cyclic codes, as a special type of constacyclic codes, have been extensively studied due to their favorable theoretical and mathematical properties. Very recently, by using the derivative of the Mattson-Solomon polynomials, Huang and Zhang (IEEE Trans Inf Theor 70(4):2395–2410, 2024) studied the cyclic derivative descendants (DDs) and linear DDs of binary extended cyclic codes and proposed the corresponding derivative decoding methods. One objective of this paper is to generalize these conclusions to q-ary extended cyclic codes with group algebra theory. It demonstrates that the cyclic DDs of a q-ary extended cyclic code are the same codes and its linear DDs are equivalent codes. In addition, we show that the relevant results can be generalized to q-ary constacyclic codes and the linear codes generated by Plotkin construction. Our conclusions reveal that the soft-decision decoding method proposed by Huang and Zhang for binary cyclic codes is also applicable to q-ary cyclic codes, q-ary constacyclic codes and the linear codes generated by Plotkin construction.

循环码作为常循环码的一种特殊类型，由于其良好的理论和数学性质而被广泛研究。最近，通过使用 Mattson-Solomon 多项式的导数，Huang 和 Zhang （IEEE Trans Inf Theor 70（4）：2395–2410， 2024） 研究了二进制扩展循环码的循环导数后代 （DD） 和线性 DD，并提出了相应的导数解码方法。本文的一个目标是将这些结论推广到具有群代数理论的 q-ary 扩展循环码。它证明了 q-ary 扩展循环码的循环 DD 是相同的代码，其线性 DD 是等效的代码。此外，我们表明相关结果可以推广到 q-ary 常循环码和 Plotkin 构造生成的线性码。我们的结论表明，Huang 和 Zhang 提出的二元循环码软决策译码方法也适用于 q-ary 循环码、q-ary 常循环码和 Plotkin 构造生成的线性码。