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REVIEW 3 major objections 5 minor 29 references

MacWilliams Theory over Zk and nu-functions over Lattices

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Lattices from ternary codes obey a new dual-counting identity, and the 1995 conjecture fails there.

desk verdict Ternary nu-function identity is a genuine new result and the counterexamples refute Solé's conjecture as stated, but the universal 'except binary' claim overreaches. read the letter →

arxiv 2504.17589 v1 pith:FUMDUST3 submitted 2025-04-24 cs.IT math.IT

Zhiyong Zheng , Fengxia Liu , Kun Tian This is my paper · ORCID
classification cs.ITmath.IT MSC 94B0511H06
keywords MacWilliamsidentityweightenumeratorcodesoverZ_knu-functionlatticesternaryConstructionA1995latticeconjecture
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 generalizes MacWilliams' identity on two fronts and then supplies a long-missing lattice analogue: a dual-counting identity for the nu-function, which enumerates lattice points by $L^1$ norm. Its main result, Theorem 4, gives a closed-form identity for the nu-function of any lattice built by Construction A from a ternary code, with a parameter relation $e^{-2\beta}=3\tanh(\alpha)/(8-5\tanh(\alpha))$ instead of the relation conjectured in 1995. The proof partitions the ternary lattice into residue classes modulo 3, identifies the dual lattice as one-third of the dual-code lattice, and converts the lattice sum into an ordinary code weight-enumerator duality. The authors also tabulate numerical failures of the 1995 conjecture for the zero-code lattices $A_k(\{0\})$ with $k=3$ to $10$, and conclude that, combined with their earlier binary-code result, the original conjecture holds only for lattices from binary codes.

What carries the argument

The nu-function $\nu_\Lambda(z)=\sum_{x\in\Lambda}z^{|x|_1}$, which counts lattice points by $L^1$ norm, is the object under study; the carrying mechanism is Construction A, the lattice $A_k(C)=\{x\in\mathbb{Z}^n: x\equiv c\pmod{k}\text{ for some }c\in C\}$. Four lemmas do the work: $\nu_{3L}(z)=\nu_L(z^3)$; $A_3(C)^*=(1/3)A_3(C^\perp)$; $\nu_{A_3(C)}(\tanh(\alpha/2))=\left(\frac{1+\tanh^3(\alpha/2)}{1-\tanh^3(\alpha/2)}\right)^n W_C\left(\frac{\tanh\alpha}{2-\tanh\alpha}\right)$, with $W_C$ the Hamming weight enumerator; and $|C|\det(A_3(C))=3^n$. The parameter relation $e^{-2\beta}=3\tanh(\alpha)/(8-5\tanh(\alpha))$ is load-bearing: it is chosen so that the ternary MacWilliams substitution maps the weight-enumerator argument correctly, and it is symmetric, so the identity holds in both dual directions.

What would settle it

Take the nonzero ternary code $C=\{00,11,22\}\subset\mathbb{Z}_3^2$, so that $\Lambda=A_3(C)=\{(a,b)\in\mathbb{Z}^2: a\equiv b\pmod{3}\}$ has index 3 and is not a binary Construction A lattice. Numerically evaluate both sides of the original 1995 identity (1.7) at $\alpha=1$, with $\beta\approx 0.136$ determined by $e^{-2\beta}=\tanh(\alpha)$: if the two sides agree, the paper's 'except binary' claim is false; if they differ, the failure extends beyond zero codes to a genuinely ternary lattice.

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Extended reading notes

Core claim

The central claim is an exact $L^1$-sphere counting identity for lattices from ternary codes. For $\Lambda=A_3(C)$, Theorem 4 states $$3^n \nu_{\Lambda^*}(\$tanh^{3}$(\$\beta$/2))=\det(\Lambda)\left[\frac{(1+\tanh(\$\beta$/2))(1-\$tanh^{3}$(\$\alpha$/2))}{(1-\tanh(\$\beta$/2))(1+\$tanh^{3}$(\$\alpha$/2))}\right]^n \nu_\Lambda(\tanh(\$\alpha$/2)),$$ where $e^{-2\beta}=3\tanh(\alpha)/(8-5\tanh(\alpha))$, a relation that is symmetric between $\alpha$ and $\beta$. The proof rests on three structural facts: multiplying a lattice by 3 changes the nu-function by $z\mapsto z^3$; the dual of $A_3(C)$ is $(1/3)A_3(C^\perp)$; and for residue classes modulo 3, the nu-function of $A_3(C)$ factors into a hyperbolic-tangent factor times the Hamming weight enumerator of $C$. The final step applies the MacWilliams identity for ternary codes, so the lattice identity is exactly as strong as the classical code identity. Against the 1995 conjecture, the paper reports numerical disagreement for the zero-code lattices $A_k(\{0\})$, $k=3$ to $10$, and asserts that the original conjecture survives only in the binary-code case.

Load-bearing premise

The paper's exclusive conclusion—that the 1995 conjecture holds only for lattices from binary codes—rests on finitely many zero-code examples $A_k(\{0\})$ for $k=3$ to $10$, with no demonstrated argument that those examples rule out all nonzero ternary codes or all other Construction A lattices; if even one such lattice satisfied the original relation, the exclusive claim would collapse.

Editorial extensions

If this is right

  • For every ternary linear code $C$, the dual lattice $A_3(C)^*$ has an exact $L^1$-sphere enumerator expressed through $C$'s weight enumerator, so those point counts can be obtained without summing over the dual lattice.
  • Because the $\alpha\leftrightarrow\beta$ relation is symmetric, Theorem 4 behaves like a true duality: applied twice, it returns the original lattice with the transformed parameter.
  • The generalized $m$-tuple identity over $\mathbb{Z}_k$ extends the earlier single-code identity to products of $m$ codes, and the complete weight-enumerator identity over $\mathbb{Z}_k[\xi]$ covers Galois rings as a special case.
  • If the negative conclusion is accepted, any future MacWilliams-type lattice identity for nonbinary alphabets must carry code-dependent parameter relations rather than the single form guessed in 1995.

Reading between the lines

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

  • The categorical 'never except binary' conclusion is stronger than the tabulated evidence: the displayed counterexamples are all zero codes $A_k(\{0\})$ for $k=3$ to $10$, and no argument shows those examples cover nonzero ternary codes or all moduli $k$. A cautious reader should treat the refutation as established for this family, and the broader statement as a supported conjecture.
  • Because $A_k(\{0\})$ is just the one-dimensional arithmetic progression $k\mathbb{Z}$, the numerical failures in the table test the parameter relation, not the code structure; testing nonzero codes in higher dimensions (such as the length-2 ternary repetition code) would show whether the obstruction is genuinely about the alphabet or only about the modulus.
  • The symmetric rational relation between $\tanh\alpha$ and $\tanh\beta$ suggests that each modulus $k$ may carry its own MacWilliams-style nu identity, with the binary case as the only one coinciding with the 1995 form; deriving such a family would be a natural next step.
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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

3 major / 5 minor

Summary. The paper has two main parts. The first extends MacWilliams theory in a coding-theoretic setting: Theorem 2 gives an m-tuple support weight enumerator identity for linear codes over Z_k, and Theorem 3 gives a complete weight enumerator MacWilliams identity for linear codes over the ring Z_k[xi]. The second part addresses Solé's 1995 conjecture (1.7) on nu-functions of lattices. Theorem 4 proves a new identity for lattices obtained by Construction A from ternary codes: for Lambda = A_3(C), the paper derives 3^n nu_{Lambda^*}(tanh^3(beta/2)) = det(Lambda)[((1+tanh(beta/2))(1-tanh^3(alpha/2)))/((1-tanh(beta/2))(1+tanh^3(alpha/2)))]^n nu_Lambda(tanh(alpha/2)), with the parameter relation e^{-2beta} = 3 tanh(alpha)/(8 - 5 tanh(alpha)). The paper then uses the one-dimensional zero-code lattices A_k({0}) = kZ for k = 3, ..., 10 as numerical counterexamples to (1.7) and claims that the conjecture never holds except for lattices associated with binary codes.

Significance. The proof of Theorem 4 is elementary, self-contained, and, as far as I checked, algebraically sound; it gives a genuine new analog of the MacWilliams identity for ternary Construction A lattices, with a parameter relation different from Solé's conjectured one. The m-tuple and complete weight enumerator identities are useful extensions of known results and are proved carefully. If the numerical counterexamples can be made rigorous, they disprove Solé's conjecture for arbitrary lattices. However, the paper's stronger 'except binary' classification claim is not established by the evidence presented, so the paper as it stands does not deliver a complete resolution of Solé's problem.

major comments (3)
  1. [Abstract; §1.2; §4] The claim that Solé's conjecture (1.7) 'never holds in the general case, except for lattices associated with a binary code' is not supported by the evidence in the manuscript. Section 4 provides counterexamples only for the one-dimensional zero codes A_k({0}) = kZ for k = 3, ..., 10. These examples show that (1.7) fails for some lattices; they do not imply that every lattice satisfying (1.7) is A_2(C) for a binary code C. To support the 'except binary' statement, the authors would need a proof of the contrapositive for all Construction A codes over Z_k with k >= 3 and for lattices outside Construction A, or at least a precise characterization of the equality cases. As written, the assertion overclaims; weakening it to 'there exist lattices for which (1.7) fails' would leave Theorem 4 intact.
  2. [§4; Table 1] The numerical counterexamples are not rigorously established. The proof of the k=5 counterexample states that inequality (4.2) can be demonstrated by choosing alpha = 1 and beta ≈ 0.136, but it does not give the numerical evaluation or an error bound. Table 1 lists decimal values for k = 3, ..., 10, and in the k = 3 row the two sides differ by 0.1000 at the displayed precision; without certified interval arithmetic or an exact algebraic comparison, the table is numerical evidence rather than a proof. Since the paper's negative answer to Solé's conjecture rests on these examples, the authors should either provide a rigorous error analysis or explicitly label these as numerical counterexamples and weaken the corresponding claims.
  3. [§1.2] The phrase 'lattices associated with a binary code' is never defined. In particular, A_3({0}) = 3Z is homothetic to A_2({0}) = 2Z, so if 'associated with a binary code' means 'up to homothety or scaling', then the k=3 row of Table 1 would not be a counterexample to the 'except binary' claim; if it means exactly Construction A_2(C), that should be stated. The classification claim is not testable until this equivalence relation is specified.
minor comments (5)
  1. [§1.2; §3] The terms 'nu-function' and 'nut function' are used interchangeably; please standardize the terminology.
  2. [§1.1; Lemma 2.1] In Lemma 2.1, the displayed formula uses the symbol a both as the summation index and as the test element, which is confusing; the intended identity should be written with separate variables.
  3. [§4] The sentence 'For any modulo k ≥ 3, it is possible to assign a value of one' is not grammatical and should be rewritten to state clearly that the authors set beta = 1 and alpha ≈ 0.136.
  4. [§4; Table 1] Table 1 should explicitly state that n = 1, C = {0}, alpha ≈ 0.136, beta = 1, and which parameter relation is used; the text confusingly mentions e^{-2alpha} = tanh(beta), while the conjecture uses e^{-2beta} = tanh(alpha). The relation is symmetric for these values, but the exposition should be precise.
  5. [References] Reference [29] is unpublished; since the binary case of the conjecture is attributed to it, the authors should clarify its status or provide a proof sketch in the paper.

Circularity Check

1 steps flagged · score 2.0 of 10

Main derivation is self-contained; only minor self-citation for the binary exceptional case.

  1. self citation load bearing [Abstract and Section 1.2, after Theorem 4]
    "In [29], we demonstrated that the aforementioned conjecture holds true for a lattice associated with a binary code C. However, in general, the following result differs significantly from the aforementioned conjecture."

    The 'except for the lattices associated with a binary code' clause in the abstract is supported only by the authors' own prior work [29], rather than being re-derived here. This is a self-citation used to justify a boundary case of the universal classification. It is not equation-level circularity: Theorem 4 and the zero-code counterexamples are proved independently from the MacWilliams identity, finite Fourier transforms, and Lemmas 3.1-3.4. The citation is load-bearing only for the binary exceptional clause, not for the main derivation.

full rationale

Theorem 4 is derived self-containedly: Lemma 3.1 gives nu_{3L}(z)=nu_L(z^3), Lemma 3.2 identifies A_3(C)^* with (1/3)A_3(C^\perp), Lemma 3.3 expresses nu_{A_3(C)}(tanh(alpha/2)) through W_C, and the proof then applies the standard q=3 MacWilliams identity (3.5) together with Lemma 3.4, |C|det(A_3(C))=3^n. There are no fitted parameters renamed as predictions, and no displayed equation is assumed equal to the target identity. The Section 4 counterexamples are concrete evaluations showing that conjecture (1.7) fails for some lattices, which is sufficient to refute the unqualified conjecture. The only circularity-adjacent element is the 'except binary' clause imported from the authors' [29]; that is a support/scope weakness rather than a circular derivation. The further exclusivity claim, that no other lattices can satisfy the conjecture, is not established by the zero-code examples, but that is an overclaim for correctness analysis, not a circularity of the proof.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central ternary formula is derived from standard generating function identities and the ternary MacWilliams identity, so the axioms are mostly standard mathematics. The paper adds one domain assumption about the freeness of Z_k[ξ] and one ad hoc assumption about the sufficiency of numerical counterexamples. No free parameters are fitted and no new entities are introduced.

assumptions (5)
  • standard math Finite Fourier transform and Poisson summation over finite abelian groups give MacWilliams identities for any finite ring's additive group.
    Invoked in the proofs of Theorems 2 and 3 via Lemma 2.3 and the factorization of the complete weight enumerator Fourier transform.
  • domain assumption The ring R=Z_k[ξ] is a free Z_k-module with basis 1,ξ,...,ξ^{t-1}; equivalently ξ is a root of a monic irreducible polynomial of degree t.
    The definitions of u(x_i), τ(x_i), and the size |R|=k^t assume unique base-k coefficient representation; the paper only says 'root of an irreducible polynomial' without stating monicity.
  • standard math Generating function identities for residue classes: ν_{3Z}(z)=(1+z^3)/(1-z^3), ν_{3Z+1}(z)=ν_{3Z+2}(z)=(z+z^2)/(1-z^3) for |z|<1.
    Used in Lemma 3.3 to compute the nu-function of A3(x).
  • standard math MacWilliams identity for Hamming weight enumerators of linear codes over Z_3 (q=3).
    Used in equation (3.5) to relate W_{C^⊥} and W_C for ternary codes.
  • ad hoc to paper The numerical values in Table 1 are accurate approximations with enough separation to establish inequality.
    The counterexample section relies on numerical evaluations without error bounds; this is an unstated assumption that approximate inequality at one point constitutes a proof of disconfirmation.

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Cite this review

Pith. "Pith review of MacWilliams Theory over Zk and nu-functions over Lattices." pith.science (2026). https://pith.science/paper/FUMDUST3

@misc{pith2026250417589,
  author       = {Pith},
  title        = {Pith review of: MacWilliams Theory over Zk and nu-functions over Lattices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FUMDUST3}},
  note         = {Machine review of arXiv:2504.17589}
}
abstract

Continuing previous works on MacWilliams theory over codes and lattices, a generalization of the MacWilliams theory over $\mathbb{Z}_k$ for $m$ codes is established, and the complete weight enumerator MacWilliams identity also holds for codes over the finitely generated rings $\mathbb{Z}_k[\xi]$. In the context of lattices, the analogy of the MacWilliams identity associated with nu-function was conjectured by Sol\'{e} in 1995, and we present a new formula for nu-function over the lattices associated with a ternary code, which is rather different from the original conjecture. Furthermore, we provide many counterexamples to show that the Sol\'{e} conjecture never holds in the general case, except for the lattices associated with a binary code.

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