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arxiv: 2607.01771 · v1 · pith:XSQ3RVFKnew · submitted 2026-07-02 · 💻 cs.IT · math.IT

On the structure of constacyclic codes over finite chain rings

Vaishali Singh , Sucheta Dutt , Ridhima Thakral This is my paper

Pith reviewed 2026-07-03 05:25 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords constacyclic codesfinite chain ringsminimum degree polynomialsMHDR codesMDS codestorsion codesresidue fieldcode generators
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The pith

Constacyclic codes of any length over finite chain rings have explicit minimal generators built from minimum-degree polynomials.

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

The paper constructs an explicit set of generators for any λ-constacyclic code of length ℓ over a finite chain ring using minimum-degree polynomials from the polynomial ring modulo x^ℓ - λ. This set has the smallest possible size and yields a minimal spanning set for the code. From this they show that the rank of the code is ℓ minus the degree of its lowest-degree generator polynomial. They also give conditions, in terms of the torsion code over the residue field, for the code to be maximum Hamming distance with respect to rank or to be maximum distance separable, and find the precise degrees that make it MHDR.

Core claim

We provide an explicit construction for generators of a λ-constacyclic code C of arbitrary length ℓ over a finite chain ring R in terms of certain minimum degree polynomials of the ring R[x]/⟨x^ℓ−λ⟩. Moreover, the proposed construction achieves the minimum possible number of generators. We prove certain properties of this set of generators, using which we obtain a minimal spanning set of C. We also obtain that the rank of C is ℓ−n0, where n0 is the degree of the minimal degree polynomial in C. Finally, we derive necessary and sufficient conditions under which an arbitrary length λ-constacyclic code C over R is Maximum Hamming Distance with respect to Rank (MHDR) as well as Maximum Distance S

What carries the argument

minimum degree polynomials in the quotient ring R[x]/⟨x^ℓ−λ⟩ that generate the ideal corresponding to the constacyclic code

If this is right

  • The generators provide a minimal spanning set for the code.
  • The rank of any such code is exactly ℓ minus n0, the degree of its minimal degree generator polynomial.
  • The code is MHDR or MDS if and only if its torsion code over the residue field satisfies the corresponding conditions.
  • The precise values of n0 determine exactly when the code is MHDR.

Where Pith is reading between the lines

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

  • The reduction to the torsion code over the residue field turns the search for good constacyclic codes over chain rings into a problem about ordinary linear codes over finite fields.
  • The explicit minimal generators may allow direct computation of code parameters without enumerating all possible generator polynomials.
  • The method supplies a uniform way to classify constacyclic codes by the degrees of their lowest-degree generators across all chain rings.

Load-bearing premise

The quotient ring R[x]/⟨x^ℓ−λ⟩ admits a well-defined notion of minimum-degree polynomials that generate the ideal corresponding to any constacyclic code, and the torsion code over the residue field Fq faithfully captures the distance properties needed for the MHDR/MDS characterizations.

What would settle it

A concrete λ-constacyclic code over a finite chain ring whose minimal generating set cannot be obtained from minimum-degree polynomials in the quotient ring, or whose MHDR status contradicts the condition on its torsion code over Fq.

read the original abstract

In the present paper, we provide an explicit construction for generators of a $\lambda$-constacyclic code $\mathcal{C}$ of arbitrary length $\ell$ over a finite chain ring(FCR) $\mathcal{R}$ in terms of certain minimum degree polynomials of the ring $\mathcal{R}[x]/ \langle x^{\ell}-\lambda \rangle$. Moreover, the proposed construction achieves the minimum possible number of generators. We prove certain properties of this set of generators, using which we obtain a minimal spanning set of $\mathcal{C}$. We also obtain that the rank of $\mathcal{C}$ is $\ell-n_0$, where $n_0$ is the degree of the minimal degree polynomial in $\mathcal{C}$. Finally, we derive necessary and sufficient conditions under which an arbitrary length $\lambda$-constacyclic code $\mathcal{C}$ over $\mathcal{R}$ is Maximum Hamming Distance with respect to Rank(MHDR) as well as Maximum Distance Separable(MDS) in terms of a torsion code of $\mathcal{C}$ over the residue field $\mathbb{F}_q$ of $\mathcal{R}$. We further determine the exact values for $n_0$ for which $\mathcal{C}$ over $\mathcal{R}$ is MHDR.

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.

Referee Report

0 major / 2 minor

Summary. The paper claims to give an explicit construction of generators for any λ-constacyclic code C of arbitrary length ℓ over a finite chain ring R, expressed via minimum-degree polynomials in the quotient ring R[x]/(x^ℓ−λ). The construction is asserted to use the minimal number of generators; the authors derive a minimal spanning set, prove that the rank of C equals ℓ−n0 (n0 the degree of the minimal-degree polynomial), and obtain necessary-and-sufficient conditions for C to be MHDR or MDS in terms of its torsion code over the residue field Fq, together with exact values of n0 that make C MHDR.

Significance. If the stated construction and characterizations hold, the work supplies a concrete, minimal-generator description of constacyclic codes over chain rings together with distance criteria controlled by the residue-field torsion code. Such explicit structure results are useful for constructing and classifying codes with prescribed distance properties over rings.

minor comments (2)
  1. [Abstract / §3] The abstract states that the construction works for arbitrary ℓ, but the manuscript should explicitly confirm that the minimum-degree polynomial is always well-defined in the quotient ring for every ideal (i.e., that the division algorithm or Gröbner-basis analogue holds without additional assumptions on λ or the chain ring).
  2. [§4] Notation for the torsion code and the precise definition of MHDR should be introduced before the characterization theorems; readers familiar with linear codes over rings may otherwise need to infer the meaning of “torsion code” from context.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our work and the recommendation of minor revision. No specific major comments were provided in the report, so we have no points to address point-by-point. We are pleased that the referee recognizes the utility of the explicit minimal-generator construction and the MHDR/MDS characterizations.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via standard ideal theory

full rationale

The paper presents an explicit generator construction for λ-constacyclic codes over finite chain rings using minimum-degree polynomials in the quotient ring R[x]/(x^ℓ−λ), derives the rank formula ℓ−n0 directly from the degree of the minimal polynomial, and obtains MHDR/MDS criteria via the torsion code over Fq. These steps rely on module-theoretic properties of ideals in chain rings and standard torsion-code distance relations, without reducing any claimed result to a fitted parameter, self-referential definition, or load-bearing self-citation. The construction is stated as achieving the minimal number of generators by explicit enumeration of the minimal-degree set, which is independently verifiable from the ring structure rather than by construction from the target code itself. No step equates a prediction to its input data or imports uniqueness via prior author work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the standard algebraic theory of finite chain rings and their polynomial quotient rings; no free parameters, invented entities, or ad-hoc axioms beyond domain-standard facts are introduced.

axioms (1)
  • domain assumption Finite chain rings possess a unique maximal ideal whose quotient is the residue field Fq, and the quotient ring R[x]/⟨x^ℓ−λ⟩ is a principal ideal ring or admits a well-behaved generator theory for constacyclic ideals.
    Invoked throughout the construction and distance characterizations; standard in the literature on codes over chain rings.

pith-pipeline@v0.9.1-grok · 5753 in / 1297 out tokens · 26171 ms · 2026-07-03T05:25:05.925216+00:00 · methodology

discussion (0)

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Reference graph

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