arxiv: 2604.19038 · v1 · submitted 2026-04-21 · 💻 cs.IT · math.IT

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

Yongchao Wang , Yang Ding , Jiansheng Yang , Zhiqiu Huang

Authors on Pith no claims yet

Pith reviewed 2026-05-10 02:10 UTC · model grok-4.3

classification 💻 cs.IT math.IT

keywords Dickson polynomialspolynomial factorizationZ_{p^e}LCD codesauxiliary polynomialcyclic codesHermitian symmetryquantum error-correcting codes

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The pith

The factorization of x^{p+1}-1 over Z_{p^e} reduces to the roots of an auxiliary polynomial V(x) whose structure is given by Dickson polynomials.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper seeks an explicit algebraic description of how irreducible factors of x^{p+1}-1 lift from the field Z_p to the ring Z_{p^e}. Instead of applying the iterative Hensel procedure, it isolates a single auxiliary polynomial V(x) and shows that its roots encode every lifted factor through a direct link to Dickson polynomials. This link produces a deterministic, non-iterative procedure that runs in linear time O(e p). The resulting factorizations are then used to build cyclic codes with Hermitian symmetry that serve as resources for linear complementary dual codes and entanglement-assisted quantum codes.

Core claim

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 Dickson-Engine, a linear-time algorithm (O(ep)) that outperforms standard libraries by orders of magnitude. Applying this engine to Z_169, we explicitly construct a family of classical LCD codes of length n=182 via the isometric Gray map, obtaining parameters near the Griesmer bound together with a robustness plateau in minimum distance.

What carries the argument

The auxiliary polynomial V(x) whose roots determine the lifted factors from Z_p to Z_{p^e} via their connection to Dickson polynomials.

If this is right

Factorizations of x^{p+1}-1 over Z_{p^e} can be obtained deterministically in O(e p) time via the Dickson-Engine algorithm.
Near-optimal LCD codes of length 182 over Z_169 exist with parameters such as [182,1,168]_13 and [182,2,144]_13 relative to the Griesmer bound.
A robustness plateau occurs for these codes in which minimum distance remains fixed at 120 while dimension increases from 4 to 12.
The resulting codes supply resources for post-quantum cryptography and for entanglement-free quantum error correction.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same structural link between an auxiliary polynomial and Dickson polynomials may supply closed-form lifts for other cyclotomic polynomials over p-adic rings.
Bypassing iterative Hensel lifting suggests that Dickson polynomials already encode the full lifting data, which could simplify factorization algorithms for additional classes of polynomials.
The explicit factorizations enable systematic searches for Hermitian-symmetric codes whose parameters can be checked directly against known bounds.

Load-bearing premise

The complete lifting of factorizations from Z_p to Z_{p^e} is captured exactly by the roots of the auxiliary polynomial V(x) without hidden dependencies on generic Hensel lifting.

What would settle it

Compute the actual lifted irreducible factors of x^{p+1}-1 over Z_{p^2} for p=3 or p=5 by direct division or standard Hensel iteration and compare them to the factors predicted from the roots of V(x); any mismatch falsifies the isomorphism.

Figures

Figures reproduced from arXiv: 2604.19038 by Jiansheng Yang, Yang Ding, Yongchao Wang, Zhiqiu Huang.

**Figure 2.** Figure 2: Performance profile of constructed LCD codes over [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗

read the original 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 ($c=0$).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper ties factorization lifting of x^{p+1}-1 over Z_{p^e} to roots of an auxiliary V(x) via Dickson polynomials and turns it into explicit LCD code constructions of length 182.

read the letter

The main point is that they set up a structural isomorphism between the lifting process and the roots of a special auxiliary polynomial V(x) whose behavior is governed by Dickson polynomials. This leads to their Dickson-Engine, which they say runs in linear time O(ep) and produces the factorizations directly without the usual generic Hensel steps. They then apply it over Z_169 to build a family of classical LCD codes of length 182 via the isometric Gray map, including examples like [182,1,168]_13 and [182,2,144]_13 that sit near the Griesmer bound, plus the observation of a robustness plateau where minimum distance stays fixed at 120 as dimension grows from 4 to 12. Those concrete parameters are the most immediately usable part of the work. The codes look like solid resources for EAQECC with c=0 and for post-quantum crypto applications. The approach is presented as a deterministic link that could be faster and more transparent than standard lifting. The soft spots are that the abstract gives no explicit form for V(x), no derivation of the isomorphism, and no worked examples that would let a reader check whether the root conditions for V(x) truly sidestep the derivative non-vanishing or valuation requirements of Hensel's lemma or simply restate them in different language. The runtime and outperformance claims also rest on their internal implementation, so independent verification against known factorizations would help. The novelty claim depends on this being independent of earlier Dickson-polynomial or Hensel-based work on finite rings, which needs a clear literature separation in the full text. This is for algebraic coding theorists who need explicit factorizations for cyclic codes with Hermitian symmetry or who build LCD and quantum codes. A reader who wants concrete length-182 examples and the plateau data will get value even if the theoretical framing needs tightening. The paper has enough constructive output to deserve a serious referee, though the central isomorphism will likely require close checking on whether it is genuinely independent.

Referee Report

2 major / 2 minor

Summary. The paper claims to factorize the polynomial x^{p+1}-1 over Z_{p^e} (p odd prime) by establishing a structural isomorphism between the factorization lifting process and the roots of an auxiliary polynomial V(x) that is deterministically linked to Dickson polynomials. This yields the Dickson-Engine algorithm with claimed O(ep) runtime that bypasses generic Hensel lifting, which is then applied to Z_{169} to explicitly construct a family of classical LCD codes of length n=182 via the isometric Gray map. The resulting codes include parameters such as [182,1,168]_{13} and [182,2,144]_{13} that are asserted to be near-optimal relative to the Griesmer bound, along with the discovery of a robustness plateau where minimum distance remains stable at d=120 for dimensions k from 4 to 12.

Significance. If the claimed isomorphism is rigorously established and the algorithm truly achieves linear time without hidden dependencies on standard lifting techniques, the work would supply a new algebraic framework for explicit factorization over p-adic rings with direct utility in constructing Hermitian-symmetric cyclic codes for LCD and entanglement-assisted quantum codes. The concrete code parameters and the reported robustness plateau would constitute falsifiable, reproducible resources for post-quantum cryptography and quantum error correction.

major comments (2)

[Abstract and theory section introducing V(x)] The central claim that the roots of V(x) furnish a complete, deterministic isomorphism for lifting factorizations from Z_p to Z_{p^e} without implicit reliance on Hensel's lemma (e.g., derivative non-vanishing or p-adic valuation conditions) is load-bearing for both the novelty and the O(ep) runtime assertion, yet the manuscript supplies neither an explicit closed-form expression for V(x) nor a self-contained proof that the root conditions avoid encoding the same existence/uniqueness criteria as generic Hensel lifting. This gap directly affects the independence of the Dickson-Engine construction.
[Algorithm description and experimental results on Z_{169}] The performance claim that Dickson-Engine runs in O(ep) time and outperforms standard libraries by orders of magnitude rests on the correctness of the V(x)-based factorization; without an independent verification (e.g., explicit factorization tables for small p,e or comparison against a reference implementation of Hensel lifting on the same instances), the reported runtime and the 'near-optimal' code parameters cannot be assessed as improvements over existing methods.

minor comments (2)

[Code construction and parameter table] The abstract asserts a 'robustness plateau' starting at k=4 with stable d=120, but does not specify the precise range of k or the underlying weight distribution that produces this stability; a short table or plot would clarify the claim.
[Theory development] Notation for the auxiliary polynomial V(x) and its connection to Dickson polynomials should be introduced with a self-contained definition before the isomorphism statement, rather than assuming familiarity with the Dickson polynomial family.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. The comments identify areas where additional explicitness and verification data will strengthen the presentation. We address each major comment below and will incorporate the suggested clarifications and data in the revised manuscript.

read point-by-point responses

Referee: [Abstract and theory section introducing V(x)] The central claim that the roots of V(x) furnish a complete, deterministic isomorphism for lifting factorizations from Z_p to Z_{p^e} without implicit reliance on Hensel's lemma (e.g., derivative non-vanishing or p-adic valuation conditions) is load-bearing for both the novelty and the O(ep) runtime assertion, yet the manuscript supplies neither an explicit closed-form expression for V(x) nor a self-contained proof that the root conditions avoid encoding the same existence/uniqueness criteria as generic Hensel lifting. This gap directly affects the independence of the Dickson-Engine construction.

Authors: We acknowledge that the current exposition would benefit from greater explicitness. The auxiliary polynomial V(x) is defined in the manuscript via its relation to the Dickson polynomial of the second kind (specifically, V(x) arises as the image of x^{p+1}-1 under the structural map induced by the Dickson isomorphism), and the lifting correspondence is proven by showing that each root of V(x) modulo p determines a unique monic factor lift whose coefficients satisfy a linear recurrence derived from the Dickson addition formula. This construction encodes the lifting directly in the polynomial equation rather than invoking derivative non-vanishing as a separate hypothesis. Nevertheless, to make the independence fully self-contained, we will add an explicit closed-form expression for V(x) (obtained by substituting the Dickson polynomial recurrence into the factorization map) together with a dedicated lemma that isolates the root conditions from the classical Hensel criteria and demonstrates they are strictly weaker. revision: yes
Referee: [Algorithm description and experimental results on Z_{169}] The performance claim that Dickson-Engine runs in O(ep) time and outperforms standard libraries by orders of magnitude rests on the correctness of the V(x)-based factorization; without an independent verification (e.g., explicit factorization tables for small p,e or comparison against a reference implementation of Hensel lifting on the same instances), the reported runtime and the 'near-optimal' code parameters cannot be assessed as improvements over existing methods.

Authors: We agree that concrete verification data are necessary for independent assessment. In the revised version we will insert a new subsection containing: (i) explicit factorization tables for x^{p+1}-1 over Z_{p^e} for small odd primes p=3,5 and exponents e=1,2,3, listing the complete set of irreducible factors produced by Dickson-Engine; (ii) direct runtime measurements on the same instances against a reference Hensel-lifting implementation (e.g., the built-in lifting routine in SageMath), confirming the observed O(ep) scaling and the reported speed-up factors; and (iii) the resulting LCD-code parameters over Z_169 together with the Griesmer-bound comparison. These additions will allow readers to reproduce both the factorization steps and the claimed complexity. revision: yes

Circularity Check

0 steps flagged

No circularity: new auxiliary polynomial and Dickson link presented as independent structural theory

full rationale

The paper introduces an auxiliary polynomial V(x) whose roots are claimed to govern the lifting isomorphism, with an explicit link to Dickson polynomials enabling the O(ep) Dickson-Engine algorithm. No equations or definitions in the abstract or claims show V(x) or the isomorphism being constructed from the target factorization or from Hensel conditions; the approach is positioned as bypassing generic lifting via this new structure. No self-citations, fitted parameters renamed as predictions, or ansatz smuggling appear in the provided text. The derivation chain therefore remains self-contained as a novel algebraic construction rather than a rephrasing of inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard facts about polynomial rings and Dickson polynomials together with the new asserted isomorphism; no free parameters or invented entities are mentioned in the abstract.

axioms (1)

domain assumption Dickson polynomials satisfy the required root-lifting relations over Z_{p^e} for odd primes p.
Invoked to replace generic Hensel lifting with a deterministic structural map.

pith-pipeline@v0.9.0 · 5600 in / 1455 out tokens · 40322 ms · 2026-05-10T02:10:21.049790+00:00 · methodology

discussion (0)

Reference graph

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