发表机构
School of Mathematical Sciences, Guizhou Normal University; College of Science, North China University of Technology（贵州师范大学数学科学学院; 华北理工大学理学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出常循环码的支撑本原分解，给出支撑周期的系数与根两种等价描述，并应用于汉明距离、权重枚举器及算术Singleton界等编码理论性质。
AI 中文摘要
设 $\mathcal C=(f)$ 为 $\mathbb F_q$ 上的 $\lambda$-常循环码，其中 $f(X)$ 是 $X^N-\lambda$ 的非常数项非零的首一因子。我们引入 $f(X)$ 的支撑周期 $\operatorname{sp}(f)$，并将其支撑本原核 $f_{\mathrm{sp}}(X)$ 定义为满足 $f(X)=f_{\mathrm{sp}}(X^s)$ 的唯一支撑本原多项式，其中 $s=\operatorname{sp}(f)$。我们证明该多项式关系诱导一个保持汉明权重的线性同构 $\mathcal C\cong\mathcal C_{\mathrm{sp}}^{\\, s}$，其中 $\mathcal C_{\mathrm{sp}}$ 是 $\mathcal C$ 的支撑本原核，并证明该分解是码的内在性质。我们给出支撑周期的两种等价描述：一种基于系数，另一种基于根。在重根情形下，后者由 $p$-adic 结构和定义函数的稳定子决定；在单根情形下，它由定义集合的最粗多重等差分表示决定。随后我们推导出汉明距离、权重枚举器、覆盖半径和欧几里得对偶方面的编码理论推论。特别地，单根常循环码的算术 Singleton 界被等同于其支撑本原核的经典 Singleton 界。最后，我们将该分解应用于具有可约生成多项式的循环码，并获得其算术 Singleton 值的界。
英文摘要
Let $\mathcal C=(f)$ be a $λ$-constacyclic code over $\mathbb F_q$, where $f(X)$ is a monic factor of $X^N-λ$ with nonzero constant term. We introduce the support period $\operatorname{sp}(f)$ of $f(X)$, and define its support-primitive core $f_{\mathrm{sp}}(X)$ as the unique support-primitive polynomial satisfying $f(X)=f_{\mathrm{sp}}(X^s)$, where $s=\operatorname{sp}(f)$. We show that this polynomial relation induces a Hamming-weight-preserving linear isomorphism $\mathcal C\cong\mathcal C_{\mathrm{sp}}^{\, s}$, where $\mathcal C_{\mathrm{sp}}$ is the support-primitive core of $\mathcal C$, and prove that this decomposition is intrinsic to the code. We give two equivalent descriptions of the support period: a coefficient-based one and a roots-based one. In the repeated-root case, the latter is determined by the $p$-adic structure and the stabilizer of the defining function, while in the simple-root case it is determined by the coarsest multiple equal-difference representation of the defining set. We then derive coding-theoretic consequences for the Hamming distance, weight enumerator, covering radius, and Euclidean duality. In particular, the arithmetic Singleton bound of a simple-root constacyclic code is identified with the classical Singleton bound of its support-primitive core. Finally, we apply the decomposition to cyclic codes with reducible generator polynomials and obtain bounds for their arithmetic Singleton values.
