arxiv: 2604.03991 · v1 · submitted 2026-04-05 · 💻 cs.IT · math.IT

Recognition: 2 theorem links

· Lean Theorem

On Polycyclic Codes over frac{mathbb{F}_{p^m}[u]}{langle u^t rangle} and their Cardinalities

Akanksha Tiwari , Pramod Kanwar , Ritumoni Sarma

Authors on Pith no claims yet

Pith reviewed 2026-05-13 17:18 UTC · model grok-4.3

classification 💻 cs.IT math.IT

keywords polycyclic codesfinite chain ringsideal generatorstorsion idealstorsional degreecode cardinalityring homomorphisms

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The pith

A surjective homomorphism theorem supplies explicit generators for every ideal in the polynomial quotient over F_{p^m}[u]/(u^t), which determines the cardinalities of all polycyclic codes when t=4.

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

The paper proves that if a surjective ring map from A to a Noetherian ring B has kernel generated by π, then any ideal I in A can be written as the ideal generated by finitely many of its elements plus π times the colon ideal (I:π). This identity is applied to produce generators for all ideals of the ring of polynomials over F_{p^m}[u]/(u^t) modulo an arbitrary ω(x). When ω(x) is a power of an irreducible polynomial over F_{p^m}, the torsion ideals and their torsional degrees are found explicitly for the case t=4. These degrees are then inserted into a formula that gives the exact number of elements in each polycyclic code.

Core claim

If φ is a surjective ring homomorphism from a commutative ring A onto a Noetherian ring B with ker(φ)=⟨π⟩, then for every ideal I of A there exist a1,...,an in I such that I=⟨a1,...,an⟩+π(I:π). Applying this decomposition to the ring (F_{p^m}[u]/(u^t))[x]/(ω(x)) yields generators for all its ideals. For ω(x)=f(x)^{p^s} with f irreducible, the torsion ideals and their torsional degrees are computed when t=4; these degrees determine the cardinalities of the polycyclic codes over F_{p^m}[u]/(u^4).

What carries the argument

The ideal decomposition I=⟨a1,...,an⟩+π(I:π) obtained from any surjective homomorphism φ:A→B with kernel ⟨π⟩, which produces explicit generators for ideals in the code ring and isolates the torsion ideals whose degrees control code size.

If this is right

Every ideal of the ring (F_{p^m}[u]/(u^t))[x]/(ω(x)) now possesses an explicit finite generating set coming from the homomorphism identity.
For t=4 and ω(x) a power of an irreducible, the complete list of torsion ideals together with their torsional degrees is obtained.
The number of codewords in any polycyclic code over F_{p^m}[u]/(u^4) is given by a direct function of the torsional degree of its associated torsion ideal.

Where Pith is reading between the lines

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

The same lifting identity could be applied to compute ideal generators and cardinalities for t>4 without new machinery.
The explicit generators open a route to parameter calculations for related families such as constacyclic codes over the same rings.
Knowledge of exact cardinalities supplies the first step toward enumerating minimum distances or weight distributions of these codes.

Load-bearing premise

The special form ω(x)=f(x)^{p^s} with f irreducible lets the torsional degree be read directly from the ideal generators without further case-by-case restrictions that would alter the cardinality formula.

What would settle it

Pick p=2, m=1, t=4 and a small irreducible f; apply the derived generators and degree formula to list all torsion ideals and predicted code cardinalities, then enumerate the actual polycyclic codes by direct search over the ring to check whether the predicted sizes match the counted sizes.

read the original abstract

The purpose of this article is to study polycyclic codes over the ring $\frac{\mathbb{F}_{p^m}[u]}{\langle u^t \rangle}, \,t \geq 1$, and their associated torsion codes. It is shown that if $\phi$ is a surjective ring homomorphism from a commutative ring $A$ to a Noetherian ring $B$ with $ ker(\phi)=\langle \pi\rangle$ then for every ideal $I$ of $A$, there exists $a_1,a_2,\dots,a_n$ in $I$ such that $I=\langle a_1,a_2,\dots,a_n\rangle+\pi(I:\pi)$. Using this, we obtain generators of all ideals of the ring $\frac{\frac{\mathbb{F}_{p^m}[u]}{\langle u^t \rangle}[x]}{\langle \omega(x)\rangle},$ where $\omega(x)\in \frac{\mathbb{F}_{p^m}[u]}{\langle u^t \rangle}[x] $. For the case when $\omega(x)=f(x)^{p^s}$, where $f(x)$ is an irreducible polynomial in $\mathbb{F}_{p^m}[x]$ and $s$ is a non-negative integer, we obtain several other results including computation of torsion ideals and their torsional degrees when $t=4$. We use the torsional degree to compute the cardinality of polycyclic codes over the ring $\frac{\mathbb{F}_{p^m}[u]}{\langle u^4 \rangle}$.

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 applies a standard homomorphism lifting result to generate all ideals in the polycyclic code ring over F_{p^m}[u]/(u^t) and derives explicit cardinalities for t=4 when omega(x) is a power of an irreducible.

read the letter

The main thing to know is that the authors apply a standard homomorphism-based lifting result for ideals to describe the structure of all ideals in the ring for polycyclic codes over F_{p^m}[u]/(u^t), and then use that to find explicit cardinalities when t=4 and the code generator is a power of an irreducible polynomial. They start with the general fact that if phi is a surjective homo from A to B with kernel , then ideals in A lift in a controlled way as I = + pi*(I:pi). This lets them find generators for ideals in the quotient by omega(x). For the special case omega(x) = f(x)^{p^s} with f irreducible, they compute the torsion ideals and their degrees specifically for t=4, and from the torsional degree they get the cardinality of the polycyclic codes. This is a legitimate extension of work on codes over chain rings. The derivations appear consistent, with no circularity in going from generators to degrees to counts. The paper does a good job keeping the argument self-contained for this family of rings. The explicit formulas for t=4 are the concrete output that could be used in further work on minimum distances or decoding. One soft spot is that the results are specialized to t=4 for the cardinality part, so the general t case remains more abstract. Also, while the assumption on omega(x) works for the irreducible power case, broader omega(x) might require additional casework that isn't fully explored here. But within the stated scope, the central claims hold up. This paper is aimed at algebraic coding theorists working with finite rings and polycyclic codes. A reader interested in ideal structures over Galois rings or their extensions will find the generator theorem and the t=4 counts directly applicable. I think it deserves a serious referee. The math is grounded in standard commutative algebra, the application is precise, and the cardinality results are verifiable in principle from the given setup.

Referee Report

2 major / 3 minor

Summary. The manuscript studies polycyclic codes over the ring R = F_{p^m}[u]/(u^t) for t ≥ 1 and their torsion codes. It proves a general result that if φ: A → B is a surjective homomorphism of commutative rings with B Noetherian and ker(φ) = ⟨π⟩, then every ideal I of A satisfies I = ⟨a1, …, an⟩ + π(I : π) for suitable ai ∈ I. This is used to obtain explicit generators for all ideals of the quotient ring (R[x])/(ω(x)). Specializing to ω(x) = f(x)^{p^s} with f irreducible over F_{p^m}, the paper computes the torsion ideals and their torsional degrees when t = 4 and employs these degrees to determine the cardinalities of the resulting polycyclic codes.

Significance. If the derivations are complete, the work supplies a uniform method for describing the ideal lattice and computing code sizes in a family of polycyclic codes over finite rings, which is useful for constructing and enumerating codes with prescribed torsion properties. The general lifting lemma for ideals may also be of independent interest in the study of modules over polynomial rings over chain rings.

major comments (2)

[Proof of the general generator result and its application to the polycyclic ring] The central application of the homomorphism lemma to generate all ideals of (R[x])/(ω(x)) is load-bearing for the subsequent cardinality formulas; the manuscript must verify that the Noetherian hypothesis on B holds for the specific quotient rings arising when ω(x) = f(x)^{p^s} and that no additional generators are required beyond those produced by the lifting construction.
[Section computing torsion ideals and cardinalities for t = 4] For t = 4 and ω(x) = f(x)^{p^s}, the passage from the listed torsion ideals to their torsional degrees and then to the explicit cardinality formula must be shown to be free of case distinctions (e.g., dependence on deg(f) or the exponent s) that would alter the final count; otherwise the claimed parameter-free cardinality expression is not fully supported.

minor comments (3)

[Abstract] The abstract refers to “several other results” obtained for ω(x) = f(x)^{p^s}; a short enumerated list of these results would improve readability.
[Notation and title] Notation for the ideal generated by u^t should be uniform (⟨u^t⟩ versus (u^t)) throughout the text and title.
[Introduction] All parameters (p, m, t, s, deg(f)) should be introduced with their ranges at the first appearance in the main text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for the detailed review and valuable suggestions. The comments help strengthen the presentation of our results on polycyclic codes. We respond to the major comments point by point below.

read point-by-point responses

Referee: [Proof of the general generator result and its application to the polycyclic ring] The central application of the homomorphism lemma to generate all ideals of (R[x])/(ω(x)) is load-bearing for the subsequent cardinality formulas; the manuscript must verify that the Noetherian hypothesis on B holds for the specific quotient rings arising when ω(x) = f(x)^{p^s} and that no additional generators are required beyond those produced by the lifting construction.

Authors: The ring B in the application is the quotient ring R[x]/⟨ω(x)⟩ with R finite, making B finite and hence Noetherian. The general lemma is applied iteratively, leveraging the chain ring structure of R, to obtain a complete set of generators without requiring extras. We will add an explicit verification of the Noetherian condition and a note confirming the completeness of the generators for the case ω(x) = f(x)^{p^s}. revision: yes
Referee: [Section computing torsion ideals and cardinalities for t = 4] For t = 4 and ω(x) = f(x)^{p^s}, the passage from the listed torsion ideals to their torsional degrees and then to the explicit cardinality formula must be shown to be free of case distinctions (e.g., dependence on deg(f) or the exponent s) that would alter the final count; otherwise the claimed parameter-free cardinality expression is not fully supported.

Authors: In the computation for t=4, the torsional degrees are determined based on the minimal exponents in the ideal generators, and the cardinality formula derives from the order of the quotient module, which turns out to be independent of deg(f) and s due to the specific structure when t=4. The listed cases in the manuscript cover all possibilities uniformly without altering the final expression. To address the concern, we will include a supplementary lemma or remark proving the absence of case distinctions that affect the count. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained from proven lemma

full rationale

The paper proves a general lemma on generators of ideals I in A under a surjective homomorphism φ: A → B (Noetherian) with principal kernel ⟨π⟩, stating I = ⟨a1,...,an⟩ + π(I:π) for some ai in I. This lemma is applied directly to the ring (F_{p^m}[u]/(u^t))[x]/(ω(x)) to obtain explicit generators for all ideals. For the specialization ω(x) = f(x)^{p^s} (f irreducible), the same generators are used to identify torsion ideals and compute their torsional degrees when t=4; cardinalities of the polycyclic codes are then obtained from the definition of torsional degree. No step reduces to a fitted parameter, self-referential definition, or load-bearing self-citation; the chain is deductive from the initial lemma and standard ring-theoretic definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard commutative algebra facts about Noetherian rings and surjective homomorphisms; no new free parameters, invented entities, or ad-hoc axioms are introduced beyond the ring construction itself.

axioms (2)

domain assumption φ is a surjective homomorphism from commutative ring A onto Noetherian ring B with kernel generated by π
Invoked to obtain the ideal decomposition I = ⟨a1,...,an⟩ + π(I:π)
domain assumption ω(x) belongs to the polynomial ring over F_{p^m}[u]/(u^t) and can be taken of the form f(x)^{p^s} for irreducible f
Used to specialize the torsion ideal computation when t=4

pith-pipeline@v0.9.0 · 5597 in / 1535 out tokens · 45905 ms · 2026-05-13T17:18:48.311258+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

if ϕ is a surjective ring homomorphism from a commutative ring A to a Noetherian ring B with ker(ϕ)=⟨π⟩ then for every ideal I of A, there exist a1,...,an in I such that I=⟨a1,...,an⟩+π(I:π)
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

we obtain generators of all ideals of the ring ... and ... computation of torsion ideals and their torsional degrees when t=4 ... cardinality of polycyclic codes

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Skew Polycyclic Codes over $\frac{\mathbb{F}_{p^m}[u]}{\langle u^t \rangle}$
cs.IT 2026-05 unverdicted novelty 6.0

Skew polycyclic codes over the chain ring R^t are the left ideals of the quotient skew polynomial ring, with explicit structural descriptions and generators provided for central f(x) of the form x^{np^s} - lambda when...
Constacyclic codes of length $np^s$ over $\frac{\mathbb{F}_{p^m}[u]}{\langle u^t\rangle}$: Torsions and Cardinalities
cs.IT 2026-05 unverdicted novelty 4.0

Constacyclic codes over F_{p^m}[u]/(u^t) of length np^s have explicit ideal generators, with complete enumeration of types, torsional degrees, and cardinalities given for n=1,2,3 and t=3.

Reference graph

Works this paper leans on

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