{"id":"c5c00373-6c57-4b90-8d63-0359670dc6c6","arxiv_id":"2412.09923","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Additive codes over Eisenstein chain rings are shown to correspond exactly, with duality preserved, to Z_p^e Z_p^{e-1}-linear codes, yielding enumeration formulas and small classifications for self-orthogonal, self-dual, and complementary-dual codes.","lead":"This paper builds a bridge between two ways of describing error-correcting codes built on number rings, showing that additive codes over Eisenstein chain rings are the same objects as certain linear codes over mixed alphabets, with dual codes matching. This lets the authors count and classify self-orthogonal, self-dual, and complementary-dual codes of this type, and they produce small optimal codes under a homogeneous weight.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"ACD enumeration rests entirely on unpublished companion [25]; without that theorem, Corollary 5.1 and Section 6 classifications cannot be independently verified.","rationale":"The reader's weakest assumption identifies exactly the same load-bearing concern: the ACD enumeration is imported from an unpublished same-author manuscript. Nothing in the present paper lets a reviewer verify Theorem 5.1, and the ACD classification sections depend on it. I checked the main duality proof and found it internally consistent: the character χ_d(c) is equivalent to the Euclidean pairing after Ψ, so Theorem 3.1 itself does not appear to be the weak point. I also spot-computed the p=3, N1=N2=2 case of Theorem 5.1 and obtained 884 including the zero code, matching the paper's reported 883 nonzero ACD codes, so the formula is plausible at that parameter set. But plausibility and two Magma checks are not enough to certify the general enumeration, especially because no scripts or logs are provided and the missing reference is by the same authors. The conditional verdict is therefore appropriate; the main correspondence is likely correct, while the ACD enumeration and the completeness of the Section 6 classifications require the companion manuscript or an independent derivation.","tokens_in":61245,"tokens_out":12789,"duration_ms":138478,"concrete_test":"Obtain [25] (or have the authors append the proof of its Theorem 3.5) and independently re-derive the closed form of Theorem 5.1 from the standard generator-matrix form of R_eR_{e-1}-linear codes, checking that the exponent (N1-i)(e-1)(i+j)+(N2-j)((e-1)i+(e-2)j) is correct for lifts of LCD codes over the residue field. Then evaluate the formula at a parameter set not in the paper, e.g., e=3, p=3, N1=N2=2, and brute-force all Euclidean Z_27Z_9-LCD codes of block-length (2,2) in Magma or Sage, including the zero code; the exhaustive count must equal the formula value exactly.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central correspondence in Theorem 3.1 is clean: the character calculation in Lemma 3.1 matches the Euclidean form, so the duality-preserving bijection is credible. The real weak point is the advertised ACD enumeration. Section 5 states Theorem 5.1 by quoting Theorem 3.5 of the authors' own manuscript [25], which is listed as 'Under review' and is not reproduced or publicly available. Corollary 5.1 and the ACD classifications in Section 6 then inherit every assumption of that unstated theorem. The Magma checks in Example 5.1 validate only two small parameter sets; they cannot distinguish a correct general exponent formula from a coincidental small-parameter match. Because [25] is unavailable, an independent reader cannot audit the ACD counts or the completeness of the 203 and 61 equivalence classes. This is the most load-bearing gap in an otherwise explicit paper.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":61321,"tokens_out":9045,"duration_ms":86609,"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":[{"comment":"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.","section":"Section 5, Theorem 5.1 and Corollary 5.1"},{"comment":"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.","section":"Section 4.3, Theorem 4.7"}],"minor_comments":[{"comment":"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.","section":"Section 3, Table 1"},{"comment":"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.","section":"Section 6, classification lists"},{"comment":"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.","section":"Section 5, Theorem 5.1"},{"comment":"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.","section":"References"},{"comment":"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'.","section":"Abstract"}],"recommendation":"major_revision","confidential_remarks":"The main concern is the dependence on the unpublished companion paper [25] for the ACD enumeration and classification results. I would recommend that the editor require the authors to either prove the needed LCD formula in this manuscript or make the companion paper available and accepted before publication. The rest of the paper is sound and could be published after the ACD results are made self-contained and the induction in Section 4.3 is expanded."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper's real content is Theorem 3.1: a duality-preserving bijection between additive codes over Eisenstein chain rings and Z_{p^e}Z_{p^{e-1}}-linear codes. The proof is a direct module-isomorphism argument, and the character calculation in Lemma 3.1 checks out. This is a genuinely useful bridge—it lets generator-matrix techniques from mixed-alphabet codes flow back to additive codes, and the self-orthogonal/self-dual/ACD correspondences in Remark 3.1 are immediate consequences. The enumeration machinery for e=2 and e=3 is detailed, follows the established mass-formula program of Yadav–Sharma, and is spot-checked against Magma. The recursive lift for e>=4 is also explicitly proved in Proposition 4.3, not waved through. The Plotkin-optimal examples in Table 1 are a small but nice bonus.\n\nThe soft spot is exactly where the stress-test note points: Section 5's ACD enumeration imports Theorem 3.5 from the authors' own manuscript [25], listed as under review and not reproduced. Corollary 5.1 and the ACD classifications in Section 6 collapse if that theorem is wrong. An independent referee cannot audit that load-bearing step. This is a real reproducibility gap, and it is the only serious one. The Magma checks are reported without scripts, so the long lists of 203 and 61 classes cannot be independently regenerated—but that is a weaker concern than the missing theorem. The spot checks for e>=4 cover only tiny parameters; still, the recurrence proof is sufficiently explicit that I read it as credible rather than coincidental.\n\nWho is this for? Specialists in algebraic coding theory working on additive codes over chain rings and linear codes over mixed alphabets. If you work on quantum code constructions from self-orthogonal codes, the correspondence is worth having on file. It does not exhibit record-breaking code families, so the impact is confined to the subfield.\n\nMy verdict: the central argument holds up. The paper deserves a serious referee, but the referee should be instructed that acceptance is conditional on either a statement/proof of the needed theorem from [25] or access to that companion manuscript. Requesting Magma scripts for the classification counts would also be reasonable. I would engage with it.","headline":"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.","tokens_in":61948,"tokens_out":1488,"would_cite":true,"duration_ms":19977,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11T71","94B60"],"pacs":[],"model":"deepseek-v4-flash","headline":"Additive codes over Eisenstein chain rings are exactly mixed-alphabet linear codes, and the correspondence preserves duality.","keywords":["Eisenstein additive codes","mixed-alphabet codes","finite chain rings","character-theoretic duality","Euclidean duality","self-orthogonal and self-dual codes","ACD codes","monomial equivalence"],"falsifier":"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.","tokens_in":60951,"feed_emoji":"🔗","tokens_out":11871,"duration_ms":111166,"temperature":0.7,"pith_summary":"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.","feed_headline":"Eisenstein additive codes are exactly mixed-alphabet linear codes","feed_subtitle":"Character duals become Euclidean duals under this map, enabling enumeration of self-dual and complementary-dual codes.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"supplies Theorem 3.5, the enumeration of Euclidean LCD codes over mixed alphabets that Corollary 5.1 imports for ACD codes.","marker":"[25]"},{"why":"provides the recursive lifting method for Euclidean self-orthogonal and self-dual codes over finite chain rings that Section 4 adapts to mixed alphabets.","marker":"[34]"},{"why":"gives standard-form generator matrices and duality for linear codes over direct products of finite chain rings, the representation used for the mixed-alphabet side.","marker":"[9]"},{"why":"gives the base count of Euclidean self-orthogonal codes over the residue field that every enumeration formula sums.","marker":"[30]"},{"why":"introduces the character-theoretic dual codes for additive codes over Eisenstein chain rings, the duality notion this paper maps to Euclidean duality.","marker":"[27]"},{"why":"provides the annihilator-of-characters duality framework for codes over modules that underlies the character-theoretic dual.","marker":"[33]"},{"why":"supplies the structure theorem for finite commutative chain rings that identifies $\\mathcal{R}_e$ with the Eisenstein quotient and supports the coefficient splitting.","marker":"[28]"},{"why":"supplies the character theory of finite abelian groups used to identify the character group of the chain ring with the split alphabet.","marker":"[23]"},{"why":"provides the formulas for $L_p(n,s)$, the number of LCD codes over $\\mathbb{Z}_p$ used inside Theorem 5.1.","marker":"[35]"}],"fun_headline_variants":["Additive Eisenstein codes equal mixed-alphabet linear codes","Character duals become Euclidean duals via Eisenstein map","Duality-preserving map unifies two code families","Mixed-alphabet codes capture Eisenstein additive codes"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Additive Eisenstein codes equal mixed-alphabet linear codes","Character duals become Euclidean duals via Eisenstein map","Duality-preserving map unifies two code families","Mixed-alphabet codes capture Eisenstein additive codes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000314,"raw_usage":{"total_tokens":1857,"prompt_tokens":1093,"completion_tokens":764,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":709,"completion_tokens_details":{"reasoning_tokens":700}},"tokens_in":709,"tokens_out":764,"duration_ms":7875,"temperature":1.0,"reasoning_tokens":700,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T16:35:10.667235+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"and Sharma, A.: On Euclidean and Hermitian LCD codes ov er mixed alphabets, Under review","cited_arxiv_id":null,"evidence_quote":"supplies Theorem 3.5, the enumeration of Euclidean LCD codes over mixed alphabets that Corollary 5.1 imports for ACD codes."},{"cited_title":"and Sharma, A.: Mass formulae for Euclidean self-ort hogonal and self-dual codes over finite commutative chain rings, Discrete Math","cited_arxiv_id":null,"evidence_quote":"provides the recursive lifting method for Euclidean self-orthogonal and self-dual codes over finite chain rings that Section 4 adapts to mixed alphabets."},{"cited_title":"and Ten-Valls, R.: Linear and cyclic codes over direct product of finite chain rings, Math","cited_arxiv_id":null,"evidence_quote":"gives standard-form generator matrices and duality for linear codes over direct products of finite chain rings, the representation used for the mixed-alphabet side."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"gives the base count of Euclidean self-orthogonal codes over the residue field that every enumeration formula sums."},{"cited_title":"and ¨Ozbudak, F.: Additive cyclic codes over finite commutative chain rings , Discrete Math","cited_arxiv_id":null,"evidence_quote":"introduces the character-theoretic dual codes for additive codes over Eisenstein chain rings, the duality notion this paper maps to Euclidean duality."},{"cited_title":"A.: Foundations of linear codes defined over finite modu les: the extension theorem and the MacWilliams identities in Codes over rings, World Scientific , pp","cited_arxiv_id":null,"evidence_quote":"provides the annihilator-of-characters duality framework for codes over modules that underlies the character-theoretic dual."},{"cited_title":"R.: Finite rings with identity , Marcel Dekker, New York (1974)","cited_arxiv_id":null,"evidence_quote":"supplies the structure theorem for finite commutative chain rings that identifies $\\mathcal{R}_e$ with the Eisenstein quotient and supports the coefficient splitting."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the character theory of finite abelian groups used to identify the character group of the chain ring with the split alphabet."},{"cited_title":"and Sharma, A.: On the enumeration and classification of σ-LCD codes over finite commutative chain rings, Discrete Math","cited_arxiv_id":null,"evidence_quote":"provides the formulas for $L_p(n,s)$, the number of LCD codes over $\\mathbb{Z}_p$ used inside Theorem 5.1."}],"review_version":1}