Support-Primitive Decomposition of Constacyclic Codes over Finite Fields: Coefficients-Based and Roots-Based Descriptions

Abstract

Let $\mathcal{C}=(f)$ be a $\lambda$-constacyclic code over ${\mathbb{F}}_q$, where $f(X)$ is a monic factor of $X^N-\lambda$ with nonzero constant term. We introduce the support period $\mathrm{sp}(f)$ 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 $s=\mathrm{sp}(f)$. 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.