Primitive Idempotents and Constacyclic Codes over Finite Chain Rings

Jo\"el Kabore; Mohammed Elhassani Charkani

Primitive Idempotents and Constacyclic Codes over Finite Chain Rings

T0 review · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Idempotents give a complete structure theorem for constacyclic codes over finite chain rings and their duals.

desk verdict The idempotent structure theorems are solid but largely known; the advertised self-dual characterization is false and contradicts the paper's own example. read the letter →

arxiv 1908.07368 v1 pith:RZJEIYSX submitted 2019-08-16 cs.IT math.ITmath.RA

Mohammed Elhassani Charkani , Joël Kabore This is my paper · ORCID

classification cs.ITmath.ITmath.RA MSC 94B1594B05

keywords finitechainringconstacycliccodeprimitiveidempotentself-dualdualsimple-rootreciprocalpolynomialcyclic

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves a structure theorem for simple-root constacyclic codes over finite chain rings. When the code length $n$ is coprime to the characteristic $p$ of the residue field, $X^n-\lambda$ factors into pairwise coprime basic irreducible polynomials, and the quotient ring $R[X]/\langle X^n-\lambda\rangle$ carries a unique complete set of primitive pairwise orthogonal idempotents. Every $\lambda$-constacyclic code then decomposes uniquely as a direct sum of components $\gamma^{r_i}\theta_i R[x]$, with $\gamma$ the generator of the maximal ideal and strictly increasing exponents $r_i$; the dual code is the mirror image with exponents $t-r_i$ and each idempotent replaced by its reciprocal. This yields explicit generators, sizes, and a characterization of non-trivial self-dual constacyclic codes, all reducing to cyclic and negacyclic cases because a non-trivial self-dual constacyclic code forces $\lambda=\pm1$.

What carries the argument

The carrying mechanism is the complete set of primitive pairwise orthogonal idempotents of the quotient ring. For $g=\prod g_i$ with pairwise coprime primary factors, a B\'ezout identity $u_ig_i+v_i\hat{g}_i=1$ produces $e_i=v_i(x)\hat{g}_i(x)$, the unique idempotent with $e_iR[x]\cong R[X]/\langle g_i\rangle$, and these $e_i$ sum to $1$. The idempotents decompose the ring into local component rings, so every ideal splits componentwise. The reciprocal-polynomial map $f\mapsto f^*$ turns ideals into ideals, and the identity $\mathcal{C}^\perp=A(\mathcal{C})^*$ converts the decomposition of $\mathcal{C}$ into the mirrored decomposition of $\mathcal{C}^\perp$. The entire code and duality theory is thereby reduced to bookkeeping of $\gamma$-exponents and idempotent-reciprocal pairings.

What would settle it

Take $R=\mathbb{Z}_4$, $n=3$, $\lambda=1$ (so the hypothesis $(n,p)=1$ holds) and enumerate every ideal of $\mathbb{Z}_4[X]/\langle X^3-1\rangle$ together with its dual. The theorem predicts each ideal is a direct sum of components $\gamma^{r_i}\theta_iR[x]$ and each dual is the mirrored sum $\gamma^{t-r_i}\theta_i^*R[x]$; any ideal failing this form disproves the structure theorem.

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Extended reading notes

Core claim

The central discovery is that, under $(n,p)=1$, the quotient ring $R[x]=R[X]/\langle X^n-\lambda\rangle$ has a unique complete set $\{e_1,\ldots,e_r\}$ of primitive pairwise orthogonal idempotents, one per monic basic irreducible factor $f_i$ of $X^n-\lambda$, with $e_iR[x]\cong R[X]/\langle f_i\rangle$. Every ideal of $R[x]$, i.e. every $\lambda$-constacyclic code, is therefore a direct sum $\mathcal{C}=\oplus_{i=1}^{r}\gamma^{s_i}e_iR[x]$ for unique $s_i\in\{0,\ldots,t\}$, and grouping equal exponents gives the canonical form $\mathcal{C}=\oplus_{i=0}^{l-1}\gamma^{r_i}\theta_iR[x]$ with $0\le r_0<\cdots<r_{l-1}<t$. The dual is then $\mathcal{C}^\perp=\oplus_{i=0}^{l}\gamma^{t-r_i}\theta_i^*R[x]$, where $*$ is the reciprocal-polynomial map, and the annihilator of $\mathcal{C}$ is the same sum before applying $*$. From this, $\mathcal{C}$ is non-trivially self-dual exactly when the exponents pair as $r_i+r_j=t$ and the corresponding idempotents $\theta_i$ and $\theta_j^*$ are associated, under the pairing $i+j\equiv 0\bmod l-1$; for cyclic and negacyclic codes this becomes the concrete cyclotomic condition $q^i\not\equiv -1\bmod n$ (or modulo $2n$) for all $i$.

Load-bearing premise

The load-bearing premise is that the code length $n$ is coprime to the characteristic of the residue field, making $X^n-\lambda$ factor without repeated factors; without that assumption the primitive-idempotent decomposition is not guaranteed to hold.

Editorial extensions

If this is right

Every simple-root $\lambda$-constacyclic code has a canonical single generator $w=\sum_{i=0}^{l-1}\gamma^{r_i}\theta_i$, so membership and size $|\mathcal{C}|=|F_q|^{\sum(t-r_i)\deg g_i}$ can be read directly from the decomposition.
The dual of any such code is obtained by the mirror recipe, so finding dual codes no longer requires solving linear equations over the ring.
A non-trivial self-dual constacyclic code can exist only for $\lambda=1$ or $\lambda=-1$, so the full constacyclic self-dual classification reduces to the cyclic and negacyclic classifications.
For cyclic codes of even nilpotency index, non-trivial self-dual codes exist exactly when some primitive idempotent is not associated to its reciprocal, equivalently when $q^i\not\equiv -1\bmod n$ for all $i$; the analogous negacyclic condition is modulo $2n$.
The results unify and extend earlier structure theorems for cyclic and negacyclic codes over $\mathbb{Z}_{p^m}$ and finite chain rings to all constacyclic codes in the simple-root case.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The idempotent decomposition suggests a direct computational method: enumerate all self-dual codes by scanning subsets of cyclotomic cosets and checking reciprocal association, rather than searching the full ideal lattice.
The reciprocal pairing on idempotents may reflect a duality on the poset of components; analyzing this duality could yield a partial structure theory for repeated-root lengths, even though the present theorem stops at simple roots.
Because self-duality forces $\lambda=\pm1$, the classification of all self-dual constacyclic codes over finite chain rings is no larger than the cyclic and negacyclic families, up to the ring isomorphisms that identify $1+\gamma R$ and $-1+\gamma R$ perturbations with those two families.
The same B\'ezout-idempotent construction could extend to finite local rings that are not chain rings whenever each component ring has a known ideal lattice, replacing the single $\gamma$-chain with whatever primary ideals occur.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The paper gives a clean idempotent decomposition for constacyclic codes over finite chain rings under the simple-root condition (n,p)=1. The construction of primitive idempotents via Bezout/CRT in Section 3 is standard but clearly presented, and Theorems 4.1-4.3 on the structure of codes and duals are correct, with a sound size argument in Corollary 4.2. If the paper stopped there it would be a reasonable expository note, though most of that material is already in Dinh-Lopez-Permouth, Kanwar-Lopez-Permouth, and Batoul et al., which the authors cite.

The advertised new result, Theorem 5.1, does not hold. The proof says "we must have theta_l = 0" and then pairs indices only up to l-1. But Theorem 4.3 gives C^perp = sum_{i=0}^l gamma^{t-r_i} theta_i^* R[x] with r_l = t, so theta_l^* always appears at exponent 0. For C = C^perp, that exponent-0 component has to match an exponent-0 component of C, which means theta_0 (or some theta_i with r_i=0) must equal theta_l^* modulo units. The condition "i+j ≡ 0 mod l-1" ignores this term. In the l=2, t=2, r0=0, r1=1 case, Theorem 5.1 demands 2r0 = t and theta_0 ~ theta_0^*, which is false, yet the paper's own Example 1 exhibits a self-dual code of that form. So the central characterization of non-trivial self-dual codes is unsupported and contradicted by the authors' example.

Other soft spots: Theorems 5.3 and 6.2 are stated without proof, just "similar to" earlier arguments. The example mixes Z72 and Z49 carelessly; it is evidently about Z49. The paper also admits it is standardizing known results, so the novelty is low.

Bottom line: the structure part is fine and might be useful to someone needing a self-contained account. The self-dual part is broken and should not be cited. I would not send this to a serious referee as is; it needs major correction or reduction to the expository sections.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the decomposition and dual formulas are derived from factorization and idempotent construction, with external results properly cited.

full rationale

The central derivation is self-contained: Theorem 3.1 constructs the primitive orthogonal idempotents of R[x]/<g> from the unique factorization of g into monic primary pairwise-coprime factors, and Theorem 4.1 uses the known chain-ring ideal structure to write every constacyclic code as C = ⊕ γ^{s_i} e_i R[x]. Theorem 4.2 then regroups equal exponents, defining θ_i as sums of primitive idempotents and θ_l as the complementary idempotent; this is an explicit structural decomposition, not an assumed conclusion. Theorem 4.3 derives C⊥ from the cited relation C⊥ = A(C)^* via the annihilator A(C), cardinality counting, and the decomposition; no fitted parameter is renamed as a prediction. The self-duality discussion in Section 5 reduces existence criteria to prior results [2], [4], [7], [9], none authored by the present authors, so there is no load-bearing self-citation chain. The paper explicitly says its method 'standardize[s] the results of [2, 4, 7, 9]', and it quotes Theorem 5.4, Proposition 5.1, and Proposition 5.2 directly from those external sources. For completeness, a correctness caveat (not a circularity): in the proof of Theorem 5.1 the paper asserts 'If C is self-dual we must have θ_l = 0' without deriving it; Theorem 4.3 gives C⊥ = ⊕_{i=0}^l γ^{t-r_i} θ_i^* R[x] with r_l = t, so the exponent-zero θ_l^* component must be matched against an exponent-zero component of C rather than simply deleted. This appears to create an inconsistency with the paper's own Example 1, but it is a mathematical gap or possible error, not a circular reduction, and therefore does not affect the circularity score.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters or invented entities. The paper rests on standard structural results for finite chain rings and on prior characterizations of constacyclic codes over fields and rings, primarily from [1,2,4,7,8,12,13].

assumptions (6)

domain assumption Every regular polynomial f over a finite local ring factors as a unit times a product of regular primary pairwise-coprime polynomials (McDonald, Thm XIII.11).
Invoked in Section 3 to factor g into primary coprime factors, the basis for the idempotent construction.
domain assumption R[X]/<f> is a finite chain ring when f is monic basic irreducible (Dinh-Lopez-Permouth, Lemma 3.1).
Used in Theorem 4.1 to conclude each component eiR[x] has ideals exactly gamma^j R[x].
domain assumption (n,p)=1, so X^n - lambda factors uniquely into monic basic irreducible pairwise coprime polynomials.
Assumed at the start of Section 4; defines the simple-root scope of the whole paper.
domain assumption Over a finite field, the only constacyclic self-orthogonal codes are cyclic and negacyclic (Kai-Zhu-Tang, Prop 2.4).
Used in Proposition 4.2 to force lambda = +/-1 for non-trivial self-orthogonal constacyclic codes.
domain assumption |C||C-perp| = |R|^n for linear codes over finite chain rings.
Used in the cardinality argument of Theorem 4.3.
domain assumption There is a ring isomorphism R[X]/<X^n - 1> to R[X]/<X^n - lambda> for lambda in 1+gammaR, and similarly for negacyclic (Batoul et al., Cor 4.5).
Used in Section 4 to reduce constacyclic self-dual codes to cyclic and negacyclic cases.

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Cite this review

Pith. "Pith review of Primitive Idempotents and Constacyclic Codes over Finite Chain Rings." pith.science (2026). https://pith.science/paper/RZJEIYSX

@misc{pith2026190807368,
  author       = {Pith},
  title        = {Pith review of: Primitive Idempotents and Constacyclic Codes over Finite Chain Rings},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RZJEIYSX}},
  note         = {Machine review of arXiv:1908.07368}
}

abstract

Let $R$ be a commutative local finite ring. In this paper, we construct the complete set of pairwise orthogonal primitive idempotents of $R[X]/<g>$ where $g$ is a regular polynomial in $R[X]$. We use this set to decompose the ring $R[X]/<g>$ and to give the structure of constacyclic codes over finite chain rings. This allows us to describe generators of the dual code $\mathcal{C}^\bot$ of a constacyclic code $\mathcal{C}$ and to characterize non-trivial self-dual constacyclic codes over finite chain rings.

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Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

[1]

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Batoul, K

A. Batoul, K. Guenda, T. A. Gulliver, Some constacyclic codes ove r ﬁnite chain rings, Advances in Mathematics of Communication 10(4) , (2016); 683-694

work page 2016

[2] [2]

Batoul, K

A. Batoul, K. Guenda, T. A. Gulliver, On self-dual cyclic codes ove r ﬁnite chain rings, Des. Codes Cryptogr. 70 , (2012), 347-358

work page 2012

[3] [3]

A. R. Calderbank, N. J. A. Sloane, Modular and p-adic codes, Designs, codes and Cryptography 6 , (1995), 21-35

work page 1995

[4] [4]

H. Q. Dinh, S. R. L´ opez-Permouth, Cyclic and negacyclic codes o ver ﬁnite chain rings, IEEE Transactions on Information Theory 50 , (2004), 1728- 1744

work page 2004

[5] [5]

S. T. Dougherty, Y. H. Park, On modular cyclic codes, Finite ﬁelds and their applications 13 , (2007), 31-57

work page 2007

[6] [6]

Guenda, T

K. Guenda, T. A. Gulliver, MDS and self-dual codes over rings, Finite Fields and Their Applications 18 (6) , (2012), 1061-1075

work page 2012

[7] [7]

X. Kai, S. Zhu, Negacyclic self-dual codes over ﬁnite chain rings, Des. Codes Cryptogr. 62 , (2012), 161-174. 16 M. H. CHARKANI and J. KABORE

work page 2012

[8] [8]

X. Kai, S. Zhu, Y. Tang, Some constacyclic self-dual codes over integers modulo 2 m, Finite ﬁeld and their applications 18(2) , (2012), 258-270

work page 2012