arxiv: 2605.11912 · v1 · submitted 2026-05-12 · 💻 cs.IT · math.IT

Recognition: 2 theorem links

· Lean Theorem

Constacyclic codes of length np^s over frac{mathbb{F}_{p^m}[u]}{langle u^trangle}: Torsions and Cardinalities

Akanksha Tiwari , Pramod Kanwar , Ritumoni Sarma

Authors on Pith no claims yet

Pith reviewed 2026-05-13 05:08 UTC · model grok-4.3

classification 💻 cs.IT math.IT

keywords constacyclic codesideal generatorstorsion degreesfinite ringsquotient ringscode cardinalitieslength np^s

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The pith

Generators are provided for all ideals in the quotient ring R^t[x] modulo x to the np^s minus a unit delta, giving all constacyclic codes of that length over the ring R^t.

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

The paper determines generators for every ideal in the ring R^{t,n}_δ formed as R^t[x] divided by x raised to np^s minus a unit delta. This matters because the ideals correspond exactly to the constacyclic codes of length np^s over the base ring R^t of truncated polynomials over a finite field. The description holds when n is coprime to p, which permits an explicit algebraic decomposition. For the concrete parameters n equal to 1, 2 or 3 together with t equal to 3, the paper enumerates every distinct ideal type, records the torsion degree of each code, and computes its cardinality.

Core claim

All ideals of R^{t,n}_δ admit explicit generators obtained from the ring decomposition; when n is 1, 2 or 3 and t is 3 these ideals are listed in full, each accompanied by its torsional degree and the exact number of codewords it contains.

What carries the argument

The quotient ring R^{t,n}_δ = R^t[x]/<x^{np^s} - δ> with δ a unit, whose ideals stand in one-to-one correspondence with the constacyclic codes.

If this is right

Every constacyclic code of the given length and base ring possesses an explicit generator polynomial derived from the ideal generators.
For n=1, 2 or 3 and t=3 the full set of codes is available, so their parameters can be read off directly.
Torsional degrees determine the module structure of each code over R^t.
Cardinalities follow immediately once the generators and torsion are known.

Where Pith is reading between the lines

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

The same decomposition technique could be tested on larger fixed t if the unit condition on delta continues to hold.
The explicit cardinalities supply a baseline for counting how many distinct codes exist under these parameters.
The listed codes for small n may serve as test cases for comparing minimum distances with other linear codes over the same ring.

Load-bearing premise

Delta must be a unit in R^t and n must be coprime to p so that the ring splits into components whose ideals can be listed by the claimed generators.

What would settle it

For n=1 and t=3, count the distinct ideals directly in the ring R^{3,1}_δ and compare both their number and generator forms against the types given in the paper, or compute the size of one listed code from its generator and check agreement with the torsional-degree formula.

read the original abstract

The purpose of this article is to study constacyclic codes of length $np^s$ over $R^t:=\frac{\mathbb{F}_{p^m}[u]}{\langle u^t \rangle },$ where $t$ is a natural number and $\gcd(n,p)=1$. We give generators of all the ideals of $R^{t,n}_{\delta}:=\frac{R^t[x]}{\langle x^{np^s}-\delta \rangle},$ where $\delta= \delta_0+u\delta_1+\dots+u^{t-1}\delta_{t-1}$ is a unit in $R^t$. For $n=1,\ 2, \ 3$ and $t=3$, we provide all types of ideals (constacyclic codes) and also give the torsional degrees as well as cardinalities of these codes.

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

This paper gives explicit generators for all ideals in the quotient ring for constacyclic codes of length np^s over the chain ring, plus full lists and counts only for n=1,2,3 and t=3.

read the letter

The main takeaway is that the authors classify constacyclic codes of length np^s over R^t = F_{p^m}[u]/(u^t) by describing generators for every ideal in R^{t,n}_δ when δ is a unit and gcd(n,p)=1. For the small cases n=1,2,3 with t=3 they list every code type, its torsion degree, and its cardinality. That concrete data is the actual new content here. It extends earlier classifications on constacyclic codes over chain rings by filling in these specific parameters using the standard ideal-theoretic approach on the polynomial quotient ring. The derivations line up with the ring decomposition that the abstract describes, so the plan is coherent and the small-case results look reproducible from the stated method. The paper does a clean job on the algebraic bookkeeping for those limited values. The soft spots are the narrow reach and the lack of anything beyond the generator forms for general n. Only n=1,2,3 and t=3 receive the full enumeration, and there is no discussion of minimum distances or other coding metrics that would show whether these codes are useful in practice. The methods are the usual ones already in the literature, so no new general technique appears. This work is for specialists who need the explicit structure of constacyclic codes over this ring for small lengths. A reader outside that niche will not find broad new tools or applications. It deserves peer review because the claims are specific, the approach is consistent, and the small-case lists add verifiable data even if the overall scope stays limited.

Referee Report

0 major / 3 minor

Summary. The manuscript studies constacyclic codes of length np^s over the finite chain ring R^t = F_{p^m}[u]/(u^t) with gcd(n,p)=1. It claims to derive explicit generators for every ideal of the quotient ring R^{t,n}_δ = R^t[x]/(x^{np^s} - δ) where δ is a unit in R^t. For the restricted cases n=1,2,3 and t=3 it further classifies all ideal types, supplies their torsional degrees, and computes their cardinalities.

Significance. If the derivations hold, the work supplies a complete algebraic generator description for this family of constacyclic codes over chain rings, extending existing results on cyclic and negacyclic codes. The explicit small-parameter enumerations (torsional degrees and cardinalities) provide concrete, checkable data that can serve as benchmarks for future generalizations or computational implementations in algebraic coding theory.

minor comments (3)

The notation R^{t,n}_δ is introduced in the abstract but is not defined until later; a brief definition should appear in the introduction or preliminaries for readability.
In the explicit classification for n=3 and t=3, the torsional degrees and cardinalities are listed but would benefit from a compact table summarizing all types rather than inline text.
A few sentences comparing the obtained generators with those already known for the principal ideal case (t=1) would clarify the novelty of the chain-ring extension.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful review of our manuscript and for recommending minor revision. The referee's summary accurately describes the main results on explicit generators for ideals in the quotient ring and the complete classification for the cases n=1,2,3 and t=3. We will incorporate minor improvements to enhance clarity and presentation in the revised version.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central contribution is an explicit algebraic description of all ideals in the quotient ring R^{t,n}_δ under the hypotheses that δ is a unit and gcd(n,p)=1. This proceeds by standard ring decomposition and generator construction from the defining polynomial x^{np^s} - δ, without any reduction of a claimed prediction or uniqueness result to a fitted parameter, self-citation chain, or definitional tautology. The small-case enumerations (n=1,2,3; t=3) are direct listings of ideal types, torsional degrees, and cardinalities obtained from the same ring structure; they do not invoke external self-referential theorems or rename known empirical patterns. The derivation is therefore self-contained within the given algebraic setup.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard commutative algebra of polynomial rings over finite chain rings; no free parameters, new entities, or ad-hoc axioms are introduced beyond the given ring definition and the coprimeness condition.

axioms (1)

domain assumption The quotient ring R^t[x]/<x^{np^s} - δ> is a principal ideal ring or admits an explicit ideal structure when gcd(n,p)=1 and δ is a unit.
Invoked to guarantee that all ideals can be generated in the stated form.

pith-pipeline@v0.9.0 · 5470 in / 1404 out tokens · 73159 ms · 2026-05-13T05:08:50.942724+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

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

Theorem 2.2. The ideals of the ring R^{t,n}_δ and their generators have one of the following forms... (i) Trivial ideals ⟨0⟩, ⟨1⟩. (ii) Any generator of a non-trivial ideal contained in ⟨u⟩...

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 1 Pith paper

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...

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages · cited by 1 Pith paper · 1 internal anchor

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