arxiv: 2605.03031 · v1 · submitted 2026-05-04 · 💻 cs.IT · math.IT· math.RA

Recognition: unknown

Cyclic codes over the ring Z2[u,v](u2(1+u),v2(1+v2))

Cristina Flaut , Bianca Liana Bercea-Straton

Authors on Pith no claims yet

Pith reviewed 2026-05-08 17:29 UTC · model grok-4.3

classification 💻 cs.IT math.ITmath.RA

keywords cyclic codeslinear codesquotient ringsfinite ringscoding theorypolynomial ringsZ2error-correcting codes

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The pith

Linear and cyclic codes are defined over the ring Z2[u,v] modulo u squared times (1 plus u) and v squared times (1 plus v squared).

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

The paper sets out to define and describe linear and cyclic codes over the specific quotient ring R equals Z2[u,v] divided by the ideal generated by u squared times (1 plus u) and v squared times (1 plus v squared). A sympathetic reader would care because extending coding theory to rings beyond fields can produce codes whose algebraic properties differ from field-based ones in ways useful for error correction. The work shows that the usual notions of linearity as R-submodules and cyclicity as ideals in the polynomial ring over R apply directly here. This description makes the codes available for further study of their parameters and constructions using standard methods.

Core claim

The central claim is that linear codes over the ring R = Z2[u,v] / (u squared (1 + u), v squared (1 + v squared)) are the R-submodules of R to the n for suitable n, while cyclic codes correspond to ideals in the quotient ring R[x] / (x to the n minus 1). These objects inherit the standard algebraic structures from the ring, including the ability to use generator matrices for linear codes and generator polynomials for cyclic ones.

What carries the argument

The finite quotient ring R = Z2[u,v] / (u squared (1 + u), v squared (1 + v squared)), which acts as the scalar ring and alphabet for the codes, allowing them to be treated as modules or ideals in the usual way.

Load-bearing premise

The given ring admits the standard definitions of linear and cyclic codes without additional compatibility conditions that would invalidate the usual generator-matrix or ideal-based constructions.

What would settle it

A concrete observation that would settle the claim is whether every R-submodule of R to the n closes under addition and scalar multiplication by elements of R, or whether every ideal in R[x] / (x to the n minus 1) is invariant under cyclic shifts of coefficients.

read the original abstract

In this paper, we describe linear and cyclic codes over the ring Z2[u,v](u2(1+u),v2(1+v2)).

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 applies the usual definitions of linear and cyclic codes to one more finite ring, with no evident new results or broad reach.

read the letter

This paper is basically a routine exercise in algebraic coding theory: it takes the definitions of linear codes as R-modules and cyclic codes as ideals in R[x]/(x^n - 1) and applies them to the ring R = ℤ₂[u, v] / (u²(1 + u), v²(1 + v²)). The abstract states that the authors describe the codes over this ring, and that appears to be the full extent of the claim. The ring is finite and commutative, so the standard constructions carry over directly without extra conditions. If the full manuscript works out explicit generator matrices, lists the cyclic ideals for small lengths, or computes parameters such as minimum distance, that would be the concrete part worth noting. The work does what it sets out to do by treating this particular ring in the usual way. The main limitation is that extensions of this type appear regularly in the coding literature, and nothing in the abstract points to surprising findings, new invariants, or connections to other problems. The scope stays narrow because it targets one specific ring, which reduces the chance of wider interest. There is also no visible derivation, example, or check in the abstract, so it is difficult to judge how thorough or accurate the details turn out to be once the full text is examined. This paper would mainly interest specialists who are systematically cataloging codes over small finite rings. For most readers working in coding theory or information theory, it will not supply new tools or change existing approaches. I would not bring this to a reading group. I would not cite it unless I needed a reference for this exact ring. It deserves to go through peer review if the manuscript contains careful, verifiable calculations on the code structures, even though the overall impact remains limited.

Referee Report

1 major / 1 minor

Summary. The manuscript states that it describes linear and cyclic codes over the ring R = ℤ₂[u,v]/(u²(1+u), v²(1+v²)). Linear codes are R-submodules of R^n and cyclic codes are ideals in R[x]/(x^n-1).

Significance. Work on linear and cyclic codes over finite commutative rings can be of interest when it yields explicit generator matrices, weight enumerators, or new families with good parameters. The present manuscript supplies none of these, so its potential significance cannot be evaluated.

major comments (1)

The manuscript consists solely of the one-sentence abstract and contains no theorems, examples, generator matrices, or explicit descriptions of the codes. This absence prevents verification of the central claim.

minor comments (1)

The ring notation in the title is non-standard and should be rendered as ℤ₂[u,v]/(u²(1+u), v²(1+v²)) with proper mathematical formatting.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their comments on our manuscript. We acknowledge that the submitted version is incomplete and consists only of the abstract, which prevents verification of the claims.

read point-by-point responses

Referee: The manuscript consists solely of the one-sentence abstract and contains no theorems, examples, generator matrices, or explicit descriptions of the codes. This absence prevents verification of the central claim.

Authors: We agree that the current manuscript contains only the abstract and lacks the promised descriptions of linear and cyclic codes over the ring R = ℤ₂[u,v]/(u²(1+u), v²(1+v²)). This was due to an error during submission preparation. The revised manuscript will include explicit descriptions of the codes as R-submodules of R^n, their generator matrices, examples, and any relevant structural results to enable verification. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper claims only to describe linear and cyclic codes over the finite commutative ring R = ℤ₂[u,v]/(u²(1+u), v²(1+v²)). Standard definitions apply verbatim: linear codes are R-submodules of R^n and cyclic codes are ideals in R[x]/(x^n-1). These constructions require no additional compatibility conditions or derivations beyond the ring being commutative and finite, so the central claim is self-contained and does not reduce to any fitted parameter, self-citation chain, or definitional renaming. No equations appear in the abstract, and the provided context contains no load-bearing steps that equate a prediction to its own input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract mentions no free parameters, no additional axioms beyond the ring definition itself, and no new invented entities.

pith-pipeline@v0.9.0 · 5316 in / 1078 out tokens · 35722 ms · 2026-05-08T17:29:13.545970+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

[1]

F., MacDonald, I

[AH; 11] Al-Ashker, M., Hamoudeh, M.,Cyclic codes overZ 2 +uZ2 +u2 2Z +...+u k−1Z2, Turk J Math, 35(2011) , 737 – 749, doi:10.3906/mat-1001-71 [AM; 69]Atiyah, M. F., MacDonald, I. G.,Introduction to Commutative Algebra, Addison-Wesley Publishing Company, London,

work page doi:10.3906/mat-1001-71 2011
[2]

[CK; 99] Cazaran, J., Kelarev, A.V.,On finite principal ideal rings, Acta Math. Univ. Comenianae, 1(LXVIII)(1999), 77–84. 13 [Ga;15]Gao, J.,Some results on linear codes overF p +uF p +u 2Fp, J. Appl. Math. Comput. 47(2015), 473–485, DOI 10.1007/s12190-014-0786-1. [KW; 82] Kasch, F., Wallace, D.A.R.,Modules and Rings,Academic Press, 1982, ISBN: 0-12-400350...

work page doi:10.1007/s12190-014-0786-1 1999