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

TL;DR

This paper introduces a structural approach to factor $x^{p+1}-1$ over $Z_{p^e}$ using Dickson polynomials, leading to a fast algorithm and the construction of near-optimal LCD codes for quantum error correction.

Contribution

It establishes a deterministic link between polynomial lifting and Dickson roots, developing a linear-time algorithm and constructing new LCD codes with stable minimum distance.

Findings

Developed Dickson-Engine, a linear-time factorization algorithm.

Constructed LCD codes with parameters close to the Griesmer Bound.

Discovered a robustness plateau where minimum distance remains stable despite increasing dimension.

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 ($c=0$).