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 ($c=0$).

翻译：设 $p$ 为奇素数。多项式 $x^{p+1}-1$ 在整数剩余环 $\mathbb{Z}_{p^e}$ 上的分解对构造具有Hermitian对称性的循环码至关重要，这类码是线性互补对偶（LCD）码和纠缠辅助量子纠错码（EAQECC）的关键资源。传统上，提升分解依赖泛化的Hensel引理，掩盖了潜在的代数结构。本文在提升过程与辅助多项式 $V(x)$ 的根之间建立结构同构，揭示了其与Dickson多项式的确定性联系。基于该理论，我们开发了线性时间算法 \texttt{Dickson-Engine}（复杂度 $O(ep)$），其性能比标准库高出数个数量级。将该引擎应用于 $\mathbb{Z}_{169}$，我们通过等距Gray映射显式构造了一类长度为 $n=182$ 的经典LCD码。搜索发现，所得码参数（例如 $[182, 1, 168]_{13}$ 和 $[182, 2, 144]_{13}$）相对于理论Griesmer界是**接近最优**的。值得注意的是，我们从非平凡维数（$k=4$）开始发现一个“鲁棒性平台”，其中最小距离保持稳定（$d=120$），即使维数增至三倍（$k=4 \rightarrow 12$）。这些码为后量子密码学和无需纠缠消耗（$c=0$）的量子纠错提供了卓越资源。