Support-primitive decomposition reveals structure of constacyclic codes
Support-Primitive Decomposition of Constacyclic Codes over Finite Fields: Coefficients-Based and Roots-Based Descriptions
Information Theory
Summary
Constacyclic codes are a type of error-correcting code used in communications and storage. This paper introduces a new way to break down these codes into simpler parts using a concept called support period. The authors show how this decomposition helps understand the code’s properties like error detection strength and distance measures. They provide two ways to describe this support period and explore mathematical consequences including improved bounds on code performance.
What this means in practice
- •For coding engineers: Refine error correction code design by using support-primitive decomposition to better estimate code distances and dual codes.
- •For cryptography teams: Analyze structural properties of constacyclic codes to enhance security assumptions in code-based cryptographic schemes.
A theory result. No direct application yet.
Authors
Li Zhu, Hongfeng Wu
Abstract
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.