Explicit Factorization of $x^{p+1}-1$ over $\mathbb{Z}_{p^e}$: A Structural Approach via Dickson Polynomials

2026-04-20

AI Summary

This work investigates the explicit factorization of the polynomial $x^{p+1} - 1$ over the ring $\mathbb{Z}_{p^e}$ to support the construction of cyclic codes with Hermitian symmetry. By establishing a structural isomorphism between the Hensel lifting process and the root set of an auxiliary polynomial $V(x)$, the study reveals for the first time a deterministic connection to Dickson polynomials and introduces Dickson-Engine, a linear-time algorithm that integrates algebraic structure analysis with Dickson theory to circumvent redundant computations inherent in conventional lifting approaches. As applications, the method yields near-optimal LCD codes over $\mathbb{Z}_{169}$ approaching the Griesmer bound—such as the $[182,1,168]_{13}$ code—and enables efficient quantum error-correcting codes without entanglement resources, while uncovering a “robustness plateau” phenomenon in minimum distance across nontrivial dimensions.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Let $p$ be an odd prime. The factorization of the polynomial $x^{p+1}-1$ over the integer residue ring $\mathbb{Z}_{p^e}$ is pivotal for constructing cyclic codes with Hermitian symmetry, a critical resource for Linear Complementary Dual (LCD) codes and Entanglement-Assisted Quantum Error-Correcting Codes (EAQECC). Traditionally, lifting factorizations relies on the generic Hensel's Lemma, masking the underlying algebraic structure. In this paper, we establish a structural isomorphism between the lifting process and the roots of a special auxiliary polynomial $V(x)$, unveiling a deterministic link to Dickson polynomials. Based on this theory, we develop \texttt{Dickson-Engine}, a linear-time algorithm ($O(ep)$) that outperforms standard libraries by orders of magnitude. Applying this engine to $\mathbb{Z}_{169}$, we explicitly construct a family of classical LCD codes of length $n=182$ via the isometric Gray map. Our search reveals codes with parameters (e.g., $[182, 1, 168]_{13}$ and $[182, 2, 144]_{13}$) that are \textbf{near-optimal} with respect to the theoretical Griesmer Bound. Notably, we discover a ``robustness plateau'' starting from non-trivial dimensions ($k=4$), where the minimum distance remains stable ($d=120$) even as the dimension triples ($k=4 \rightarrow 12$). These codes provide exceptional resources for post-quantum cryptography and quantum error correction without entanglement consumption ($c=0$).

Problem

• polynomial factorization
• cyclic codes
• Hermitian symmetry
• LCD codes
• EAQECC

Innovation

• Dickson polynomials
• explicit factorization
• linear-time algorithm
• LCD codes
• quantum error correction