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On Eisenstein additive codes over chain rings and linear codes over mixed alphabets

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Pith's one-line read Additive codes over Eisenstein chain rings are exactly mixed-alphabet linear codes, and the correspondence preserves duality.

desk verdict Main correspondence is real and useful; ACD enumeration has a genuine reproducibility gap because it imports an unstated theorem from the authors' own unpublished manuscript. read the letter →

arxiv 2412.09923 v1 pith:HT6CEDWY submitted 2024-12-13 cs.IT math.IT

Leijo Jose , Anuradha Sharma This is my paper · ORCID
classification cs.ITmath.IT MSC 11T7194B60
keywords Eisensteinadditivecodesmixed-alphabetfinitechainringscharacter-theoreticdualityEuclideanself-orthogonalandself-dualACDmonomialequivalence
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

This paper establishes a one-to-one, duality-preserving correspondence between additive codes over a finite commutative chain ring $\mathcal{R}_e$ (an Eisenstein extension of a Galois ring) and linear codes over mixed alphabets $\mathbb{Z}_{p^e}\mathbb{Z}_{p^{e-1}}$. Under this correspondence, the character-theoretic dual of an additive code becomes the Euclidean dual of the corresponding mixed-alphabet code, so generator-matrix methods for mixed-alphabet codes become available for additive codes over chain rings. The authors use this to enumerate self-orthogonal and self-dual additive codes of any length when the residue characteristic is odd, and to produce a formula for complementary-dual (ACD) additive codes. They also translate monomial equivalence of additive codes into an equivalence of mixed-alphabet codes and carry out classifications for lengths 2 and 3 over two concrete chain rings. A short table of additive codes meeting the Plotkin bound for homogeneous weights suggests the class is a useful source of optimal codes.

What carries the argument

The load-bearing object is the coefficient-wise isomorphism $\Psi:\mathcal{R}_e\to \mathbb{Z}_{p^e}^{rt}\oplus \mathbb{Z}_{p^{e-1}}^{r(k-t)}$, built by writing each element of $\mathcal{R}_e$ as $a_0+\cdots+a_{t-1}y^{t-1}+a_t y^t+\cdots+a_{k-1}y^{k-1}$ with coefficients in $GR(p^e,r)$ for the first $t$ positions and in $GR(p^{e-1},r)$ for the remaining $k-t$ positions, then applying the Galois-ring coefficient map $\varphi$. Applied coordinate-wise, $\Psi$ carries additive codes to mixed-alphabet linear codes, and the character $\chi_a$ is defined so that $\chi_d(c)=1$ is equivalent to the Euclidean product $\langle \Psi(d),\Psi(c)\rangle_E=0$, which is why duality survives the correspondence. The counting arguments then use standard-form generator matrices for mixed-alphabet codes and a recursive lift from $\mathcal{R}_{\mu-2}\mathcal{R}_{\mu-3}$-codes to $\mathcal{R}_\mu\mathcal{R}_{\mu-1}$-codes.

What would settle it

For a small case such as length $N=2$ over $\mathcal{R}_2=\mathbb{Z}_4[y]/\langle y^2-2,2y\rangle$, enumerate all additive subgroups of $\mathcal{R}_2^2$ directly, compute $C^{\perp_\chi}$ from the character definition, and count those with $C\cap C^{\perp_\chi}=\{0\}$; compare the result with the claimed 113 ACD codes. Also check $\Psi(C^{\perp_\chi})=\Psi(C)^{\perp_E}$ for every such $C$, since a single mismatch would refute Theorem 3.1.

Watch

Extended reading notes

Core claim

The central claim is Theorem 3.1: a non-empty subset $C \subseteq \mathcal{R}_e^N$ is an additive code if and only if $\Psi(C)$ is a $\mathbb{Z}_{p^e}\mathbb{Z}_{p^{e-1}}$-linear code of block-length $(Nrt, Nr(k-t))$, and $\Psi(C^{\perp_\chi})=\Psi(C)^{\perp_E}$. In other words, additive codes over the Eisenstein chain ring are exactly the mixed-alphabet linear codes of that block-length, and the character-theoretic dual operation is exactly the Euclidean dual operation transported through $\Psi$. The paper then exploits this identity to construct and count self-orthogonal, self-dual, and complementary-dual additive codes by counting generator matrices of mixed-alphabet codes, and to classify small cases up to monomial equivalence.

Load-bearing premise

The ACD enumeration rests on the cited unpublished manuscript [25], whose Theorem 3.5 is used without proof to count Euclidean LCD codes over mixed alphabets; if that theorem is wrong or inapplicable, the ACD formula in Corollary 5.1 and the ACD classifications collapse, while the main correspondence Theorem 3.1 would still stand.

Editorial extensions

If this is right

  • Because $\Psi$ identifies additive codes with $\mathbb{Z}_{p^e}\mathbb{Z}_{p^{e-1}}$-linear codes, every generator-matrix construction for mixed-alphabet codes yields an additive code over $\mathcal{R}_e$ together with its character-theoretic dual.
  • Self-orthogonal, self-dual, and ACD additive codes correspond respectively to Euclidean self-orthogonal, self-dual, and LCD mixed-alphabet codes, so the paper's enumeration formulae directly enumerate these special classes of additive codes.
  • When $p$ is odd, the existence of a self-dual additive code of length $N$ is settled by parity and quadratic-residue conditions on $Nrt$ or $Nr(k-t)$, and the explicit sums in Corollaries 4.1-4.6 give the number of such codes.
  • Monomial equivalence of additive codes becomes $*$-equivalence of mixed-alphabet codes, which is how the paper obtains classifications for lengths 2 and 3 over $\mathbb{Z}_9[y]/\langle y^2-3,3y\rangle$ and length 2 over $\mathbb{Z}_4[y]/\langle y^2-2,2y\rangle$.
  • The listed codes over $\mathbb{Z}_4[y]/\langle y^2-2,2y\rangle$ meet the Plotkin bound for homogeneous weights, so additive codes over $\mathcal{R}_e$ are a concrete source of optimal codes in the homogeneous metric.

Reading between the lines

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

  • Because the correspondence uses only the coefficient-wise module splitting of $\mathcal{R}_e$, the same construction should extend to other towers $\mathcal{R}_\mu/\mathcal{R}_{\mu-1}$ and other Eisenstein polynomials; testing that extension would be a natural next step.
  • If the unpublished enumeration theorem [25] changes when it appears, the ACD counts would need revision, but the duality correspondence and the self-orthogonal/self-dual enumerations would not be affected.
  • The same $\Psi$ could be used to search larger lengths for additive codes with good homogeneous-weight parameters, since the Table 1 examples show only small instances; whether such codes beat current nonlinear codes is not established in the paper.
  • The block-diagonal $*$-equivalence notion may be useful for studying translation-invariant or propelinear structure of mixed-alphabet codes, but the paper does not pursue that direction.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper considers additive codes over the finite commutative chain ring R_e = GR(p^e,r)[y]/⟨g(y), p^{e−1}y^t⟩, where g is an Eisenstein polynomial. Its main structural result, Theorem 3.1, establishes a duality-preserving Z_{p^e}-module isomorphism between additive codes of length N over R_e and Z_{p^e}Z_{p^{e−1}}-linear codes of block-length (N rt, N r(k−t)), under which the character-theoretic dual C^{⊥χ} maps to the Euclidean dual of Ψ(C). Using the standard generator-matrix theory of mixed-alphabet chain-ring codes, the paper then derives enumeration formulae for self-orthogonal and self-dual additive codes for odd p (Sections 4.1–4.3), an enumeration formula for complementary-dual (ACD) codes (Section 5), and a translation of monomial equivalence into ∗-equivalence (Section 6). The paper also lists small additive codes over Z_4[y]/⟨y^2−2,2y⟩ attaining the Plotkin bound for homogeneous weights and classifies self-orthogonal, self-dual, and ACD codes of lengths 2 and 3 over two explicit chain rings up to monomial equivalence.

Significance. The correspondence in Theorem 3.1 is a valuable and convincing structural contribution: it is proved from first principles via the character calculation in Lemma 3.1 and equation (3.3), and it replaces the relatively unstructured category of additive codes over Eisenstein chain rings by the well-studied generator-matrix theory of mixed-alphabet linear codes. The enumeration of self-orthogonal and self-dual codes in Section 4 is self-contained for e = 2 and e = 3, with the counting formulas checked by Magma for several small parameter sets; the recursive lifting in Proposition 4.3 and the resulting closed forms for e ≥ 4 are plausible, though the induction is compressed. The paper also deserves credit for providing explicit small classifications up to monomial equivalence. However, as advertised, the ACD enumeration is not self-contained: Theorem 5.1 is quoted from the authors' own unpublished manuscript [25], so Corollary 5.1 and the Section 6 ACD classifications cannot be independently verified from the present manuscript.

major comments (2)
  1. [Section 5, Theorem 5.1 and Corollary 5.1] Theorem 5.1, the enumeration formula for Euclidean Z_{p^e}Z_{p^{e−1}}-LCD codes, is quoted from the authors' own manuscript [25], listed in the references as 'Under review', and its proof is not reproduced in this paper. Corollary 5.1 and the ACD classifications in Section 6 (items V and VI, including the counts 203 and 61) depend on this unstated theorem. Since [25] is not publicly available, the counting results for ACD codes cannot be audited by a reader. Please provide a complete proof of Theorem 5.1 within this paper, or alternatively replace Corollary 5.1 and the affected classification claims with results that are proved here.
  2. [Section 4.3, Theorem 4.7] The proof of Theorem 4.7 is the single sentence 'By repeatedly applying the recurrence relation derived in Proposition 4.3(c)...' while the assembled formula in (4.63)-(4.65) is complicated. For a result covering all e ≥ 4, the induction should be written out: state the induction hypothesis, show how k^(i) and ℓ^(j) evolve under the recurrence, and verify that the exponents Δ_e(k,ℓ) + s_e(k,ℓ) telescope to the claimed value. As written, the generalized enumeration formula is not fully verifiable by a reader.
minor comments (5)
  1. [Section 3, Table 1] The table reports 'Optimal' codes but does not display the homogeneous Plotkin bound value for each parameter set, so the optimality claim is not directly checkable from the table; please add the bound value or a reference to the computed bound.
  2. [Section 6, classification lists] Several generator-matrix listings appear to have mismatched numbers of variables: for example, in item III the matrix form '[3 0 x | 0 0 0]' is followed by six-tuples, and a matrix with six displayed entries is followed by ten-tuples. Please correct the displayed forms or explain the shorthand, since these lists are intended to be complete classification data.
  3. [Section 5, Theorem 5.1] The exponent in the displayed formula is ambiguous: 'p(N1−i)(e−1)(i+j)+(N2−j)((e−1)i+(e−2)j)' should be written with an explicit brace or parentheses as p^{(N1−i)(e−1)(i+j)+(N2−j)((e−1)i+(e−2)j)} to make clear that the whole expression is the exponent.
  4. [References] Reference [25] is listed as 'Under review' with no preprint identifier; if it remains a dependency, please provide a public version or a statement of its status.
  5. [Abstract] The abstract contains a subject-verb agreement error: 'additive codes over Re is a promising class' should read 'additive codes over Re are a promising class'.

Circularity Check

1 steps flagged · score 4.0 of 10

The main duality correspondence is derived in-paper, but the advertised ACD enumeration reduces to the authors' unpublished Theorem 3.5 in [25].

  1. self citation load bearing [Section 5, Theorem 5.1 and Corollary 5.1; Reference [25]]
    "Theorem 5.1 ([25]). ... Proof. It follows from Theorem 3.5 of Jose and Sharma [25]. ... Corollary 5.1. ... Proof. It follows by Remark 3.1 and taking N1 = N rt and N2 = N r(k − t) in Theorem 5.1."

    The ACD enumeration formula advertised in the abstract is not derived in this paper. Corollary 5.1 consists solely of substituting N1 = Nrt and N2 = Nr(k−t) into Theorem 5.1, and Theorem 5.1's proof is deferred to [25], an unpublished manuscript by the same two authors listed as 'Under review'. The Magma checks in Example 5.1 validate only the two small parameter sets (2,2) over Z4[y]/⟨y2−2,2y⟩ and Z9[y]/⟨y2−3,3y⟩; they do not prove the general formula. Consequently the ACD counts, and the Section 6 classifications V and VI that apply Theorem 5.1, reduce to an unverified self-citation rather than to an argument contained in this paper.

full rationale

No circularity was found in the central correspondence. Theorem 3.1(a) follows from Ψ being a Zpe-module isomorphism, and Theorem 3.1(b) follows from Lemma 3.1's explicit character computation, which matches the Euclidean bilinear form (2.2); this is a translation, not a definitional loop. The Section 4 enumeration formulae for Euclidean self-orthogonal and self-dual ReRe−1-linear codes are derived in-paper from external lemmas (e.g., Lemma 4.1 from [6,34]) and the recursive lifting method, with Magma checks for several parameter sets. The only load-bearing self-citation is the ACD enumeration: Corollary 5.1 imports Theorem 5.1 from the authors' own unpublished manuscript [25], listed as 'Under review'. Because Theorem 3.5 of [25] is not reproduced, the general ACD formula cannot be audited from this paper alone, and the Section 6 classifications of ACD codes inherit that dependency. This is genuine self-citation load-bearing, but it is not a case of fitted parameters being renamed as predictions or of the main duality theorem reducing to its own input; hence the score is 4, not higher.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No free parameters are fitted and no new entities are postulated. The derivations rest on standard character theory and chain-ring facts, known mass formulas (Pless, Betty-Munemasa, Yadav-Sharma), and one unpublished same-author theorem [25] used as a black box for ACD counts. The odd-characteristic assumption in Section 4 restricts the enumeration theorems to p odd.

assumptions (7)
  • standard math Re = GR(p^e,r)[y]/⟨g(y), p^{e-1}y^t⟩ is a finite commutative chain ring with maximal ideal generated by y and nilpotency index k(e-1)+t (Theorem XVII.5 of McDonald [28]).
    Foundation for all code structures; invoked in Section 2.1.
  • standard math Every R_µR_{µ-1}-linear code has a unique standard-form generator matrix (Proposition 3.2 of Borges et al. [9]).
    Needed for enumeration and dual type; invoked in Section 2.3 and throughout Section 4.
  • standard math The character group of a finite Abelian group is isomorphic to the group itself, and product characters represent direct products (Theorem 5.1 of Huppert [23]).
    Used to set up character-theoretic duality in Section 2.2 and Lemma 3.1.
  • standard math Lemma 4.1: Φ_A(B) = AB^T + BA^T is surjective with kernel size q^{s(2n-s-1)/2}, and AX = J has q^{(n-s)ℓ} solutions.
    Central counting device in Propositions 4.1, 4.2, and 4.3; cited from Betty and Munemasa [6] and Yadav and Sharma [34].
  • standard math The number of Euclidean self-orthogonal codes over F_q of length n and dimension s is given by the Pless formula (4.1).
    Base counts in all enumeration theorems; cited from Pless [30].
  • domain assumption Theorem 3.5 of Jose and Sharma [25] gives the number of Euclidean Z_{p^e}Z_{p^{e-1}}-LCD codes of block-length (N1,N2).
    The paper assumes this theorem as a black box for all ACD counts; the cited work is unpublished and by the same authors, so the assumption is not independently checkable from the preprint.
  • domain assumption Throughout Section 4, the chain ring Re is assumed to have odd characteristic, i.e., the residue field has odd prime power order q.
    Needed for Lemma 4.1 and the Pless-based counts; restricts the enumeration theorems to p odd.

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Pith. "Pith review of On Eisenstein additive codes over chain rings and linear codes over mixed alphabets." pith.science (2026). https://pith.science/paper/HT6CEDWY

@misc{pith2026241209923,
  author       = {Pith},
  title        = {Pith review of: On Eisenstein additive codes over chain rings and linear codes over mixed alphabets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HT6CEDWY}},
  note         = {Machine review of arXiv:2412.09923}
}
abstract

Let $\mathcal{R}_e=GR(p^e,r)[y]/\langle g(y),p^{e-1}y^t\rangle$ be a finite commutative chain ring, where $p$ is a prime number, $GR(p^e,r)$ is the Galois ring of characteristic $p^e$ and rank $r,$ $t$ and $k$ are positive integers satisfying $1\leq t\leq k$ when $e \geq 2,$ while $t=k$ when $e=1,$ and $g(y)=y^k+p(g_{k-1}y^{k-1}+\cdots+g_1y+g_0)\in GR(p^e,r)[y]$ is an Eisenstein polynomial with $g_0$ as a unit in $GR(p^e,r).$ In this paper, we first establish a duality-preserving 1-1 correspondence between additive codes over $\mathcal{R}_e$ and $\mathbb{Z}_{p^e}\mathbb{Z}_{p^{e-1}}$-linear codes, where the character-theoretic dual codes of additive codes over $\mathcal{R}_e$ correspond to the Euclidean dual codes of $\mathbb{Z}_{p^e}\mathbb{Z}_{p^{e-1}}$-linear codes, and vice versa. This correspondence gives rise to a method for constructing additive codes over $\mathcal{R}_e$ and their character-theoretic dual codes, as unlike additive codes over $\mathcal{R}_e,$ $\mathbb{Z}_{p^e}\mathbb{Z}_{p^{e-1}}$-linear codes can be completely described in terms of generator matrices. We also list additive codes over the chain ring $\mathbb{Z}_4[y]/\langle y^2-2,2y \rangle$ achieving the Plotkin's bound for homogeneous weights, which suggests that additive codes over $\mathcal{R}_e$ is a promising class of error-correcting codes to find optimal codes with respect to the homogeneous metric.

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