Codes over Finite Ring $\mathbb{Z}_k$, MacWilliams Identity and Theta Function

Fengxia Liu; Kun Tian; Zhiyong Zheng

REVIEW 3 major objections 3 minor 47 references

Codes over Finite Ring $\mathbb{Z}_k$, MacWilliams Identity and Theta Function

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

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Pith's one-line read This paper proves that the complete weight enumerator MacWilliams identity for linear codes over the finite ring $\mathbb{Z}_k$ holds for every positive integer $k$, extends the genus-$g$ identity beyond the even-$k$ case, and, for odd…

desk verdict Correct but modest genus-g MacWilliams result sits inside a paper whose headline claim—deriving the full complete weight enumerator MacWilliams identity from one-variable theta functions—is invalid, because the theta functions are symmetric and the algebraic independence the authors say they need is impossible. read the letter →

arxiv 2504.12604 v1 pith:NNBTGNAY submitted 2025-04-17 cs.IT cs.CRmath.IT

Zhiyong Zheng , Fengxia Liu , Kun Tian This is my paper · ORCID

classification cs.ITcs.CRmath.IT MSC 94B0511F2711T71

keywords linearcodesoverZ_kMacWilliamsidentitycompleteweightenumeratorsthetafunctionsassociatedlatticescyclotomicfieldsgenusgPoissonsummationformula

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's aim is to settle the MacWilliams identity for codes over $\mathbb{Z}_k$ for every positive integer $k$ by viewing a code through its associated lattice. It shows that the $\theta$ function of the lattice $\Gamma_C = \frac{1}{\sqrt{k}}\rho^{-1}(C)$ equals the complete weight enumerator of $C$ evaluated at $k$ auxiliary $\theta$ functions, and uses Poisson summation to turn this into complete and symmetrized weight enumerator identities. The genus-$g$ identity, previously proved only for even $k$, is stated and proved for all $k$ via a Fourier transform over $\mathbb{Z}_k^g$. For odd primes $p$, the paper proves that the $\theta$ function of the cyclotomic lattice attached to a self-orthogonal code $C \subset \mathbb{F}_p^n$ equals the Hamming-weight complete weight enumerator $W_C(\vartheta_0,\ldots,\vartheta_{p-1})$, generalizing the Lee-weight $\theta$ identity over cyclotomic fields. The paper leaves one step open: converting the $\theta$-function evaluation in Theorem 2 into the full polynomial identity requires algebraic independence of the auxiliary $\theta$ functions, which it says it has not yet proved.

What carries the argument

The load-bearing objects are the $k$ $\theta$ functions $A_j(z)=\sum_{x\in k\mathbb{Z}+j} e^{\pi i z x^2/k}$ for $j=0,\ldots,k-1$, together with the associated lattice $\Gamma_C=\frac{1}{\sqrt{k}}\rho^{-1}(C)$. Their Poisson summation transform, $A_j(-1/z)=\frac{1}{\sqrt{k}}(z/i)^{1/2}\sum_{m=0}^{k-1} e^{2\pi i j m/k}A_m(z)$, is exactly the Fourier matrix that appears in the MacWilliams identity, so the $\theta$ function of the associated lattice becomes a weighted sum of the $A_j$'s. In genus $g$, the same Fourier kernel appears as $T=(\eta^{a\cdot b})_{a,b\in\mathbb{Z}_k^g}$, and the Poisson summation formula over $\mathbb{Z}_k^{g\times n}$ carries the proof. For $k=p$ prime, the machinery moves to the cyclotomic field $K=\mathbb{Q}(\xi)$, $\xi=e^{2\pi i/p}$, with $\theta$ functions $\vartheta_j(z)=\sum_{x\in\mathfrak{B}+j} e^{2\pi i z \mathrm{Tr}_{K^+/\mathbb{Q}}(x\bar{x}/p)}$ over the principal ideal $\mathfrak{B}=\langle1-\xi\rangle$; the same factorization argument gives the complete weight enumerator identity.

What would settle it

For a fixed small $k$ (for example $k=2$ or $k=3$), compute the $q$-expansions of $A_0,\ldots,A_{k-1}$ to sufficiently high order and use linear algebra to look for a nonzero homogeneous polynomial relation among them. Finding such a relation would show that Theorem 2 does not by itself yield the polynomial complete weight enumerator MacWilliams identity; proving that no such relation exists, for instance by showing the image of $z\mapsto(A_0(z),\ldots,A_{k-1}(z))$ is Zariski dense, would complete the paper's stated open step.

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Extended reading notes

Core claim

The central discovery is a dictionary between codes and lattices that makes the MacWilliams identity a $\theta$-function statement. For any linear code $C \subset \mathbb{Z}_k^n$, with associated lattice $\Gamma_C = \frac{1}{\sqrt{k}}\rho^{-1}(C)$, the paper proves $\vartheta_{\Gamma_C}(z) = W_C(A_0(z),\ldots,A_{k-1}(z))$, where the $A_j$ are the $\theta$ series attached to the residue classes of $\mathbb{Z}$ modulo $k$ (Theorem 1). This is the starting point for Theorem 2, the genus-1 identity evaluated at the $A_j$, and for the symmetrized identity. The unrestricted claim is Theorem 3: for every positive integer $k$, the genus-$g$ complete weight enumerator satisfies $C_{C^\perp,g}(z_a) = \frac{1}{|C|^g} T\,C_{C,g}(z_a)$ with $T=(\eta^{a\cdot b})$, $\eta=e^{2\pi i/k}$, proved by Fourier transform and Poisson summation. For odd prime $p$, Theorem 4 gives $\vartheta_{\Gamma_C}(z)=W_C(\vartheta_0(z),\ldots,\vartheta_{p-1}(z))$ for $C\subset C^\perp$, with $\Gamma_C=\rho^{-1}(C)\subset D^n$ in the cyclotomic integer ring and with the $\vartheta_j$ defined by the ideal $\mathfrak{B}=\langle1-\xi\rangle$. The paper states that Theorem 2 is equivalent to the ordinary complete weight enumerator MacWilliams identity only if the functions $A_0,\ldots,A_{k-1}$ are algebraically independent, and it lists that proof as an open problem.

Load-bearing premise

The load-bearing premise is that the $k$ $\theta$ functions $A_0(z),\ldots,A_{k-1}(z)$ are algebraically independent over $\mathbb{C}$ — that is, no nonzero polynomial in $k$ variables vanishes after the substitution — so an identity that holds at these special functions is an identity of polynomials; the paper says this proof is still missing in its conclusion.

Editorial extensions

If this is right

For every positive integer $k$, the genus-$g$ complete weight enumerator MacWilliams identity holds for codes over $\mathbb{Z}_k$, removing the even-$k$ condition of the earlier genus-$g$ result.
Taking $g=1$ in Theorem 3 gives the ordinary complete weight enumerator MacWilliams identity for all $k$, and the symmetrized weight enumerator version follows by identifying $a$ with $-a$.
For a self-dual code over $\mathbb{Z}_k$, the complete weight enumerator is invariant under the rotation described in Corollary 1.
For odd primes $p$, a self-orthogonal code over $\mathbb{F}_p$ gives a cyclotomic lattice whose theta function equals its Hamming complete weight enumerator, and the same identity holds for $z\in\mathbb{H}^{p-1}$.
If the missing algebraic independence is established, Theorem 2 becomes a genuine modular-form proof of the usual MacWilliams identity.

Reading between the lines

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

The open algebraic-independence step is not needed to obtain the MacWilliams identity itself, since Theorem 3 with $g=1$ already proves the polynomial identity by Fourier analysis; the open step affects only the claim that the theta-function route recovers it.
The cyclotomic construction for primes $p$ suggests a testable extension to composite $k$ through the integers of an appropriate cyclotomic field, which the authors flag as future work; the main obstruction to check is whether the trace form remains positive definite.
If the $A_j$ turn out to be algebraically dependent, the theta-function proof could still be rescued by replacing the evaluation point with a generic point of the resulting algebraic variety, though that would require a different modular argument.
Because the matrix $T$ in Theorem 3 is the character table of the finite abelian group $\mathbb{Z}_k^g$, the identity is a Fourier-duality statement about weight distributions, which may connect the result to finite abelian harmonic analysis beyond coding theory.

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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

The paper has two genuinely correct pieces: the genus-g complete weight enumerator MacWilliams identity for all positive k (Theorem 3), proved cleanly by Fourier transform and Poisson summation, and the cyclotomic lattice theta identity (Theorem 4), which is also a direct computation. Theorem 3 is a legitimate extension of Bannai et al.'s even-k result to arbitrary k, though it is routine rather than deep. Theorem 1 is standard construction-A material, correctly stated. The citation pattern is fine, and the self-citations are contextual; the proofs do not lean on them.

The main advertised result does not survive contact with the paper. The functions A_j defined in Definition 7 satisfy A_j = A_{k-j} identically, because x \mapsto -x is a bijection from kZ+j to kZ+(k-j) and the exponent depends on x^2. So the vector (A_0(z),...,A_{k-1}(z)) always lies in the proper subspace X_j = X_{k-j}; it is not Zariski dense in C^k. That means the equality proved in Theorem 2 is only a functional identity on that subspace. You cannot replace the A_j by independent polynomial variables and conclude the polynomial complete weight enumerator MacWilliams identity. The authors themselves flag the need for algebraic independence in Section VI, but that independence is not merely unproved; it is false. The same symmetry afflicts Theorem 4: the theta functions satisfy \vartheta_j = \vartheta_{p-j}, so the theta function of the associated lattice sees only the symmetrized (Lee-type) enumerator, not the full Hamming complete weight enumerator. Calling Theorem 4 a generalization of van der Geer–Hirzebruch to complete weight enumerators overstates what the identity delivers.

There is also a minor mechanical slip: the proof of Theorem 2 invokes Proposition 2 where it needs Proposition 3.

Bottom line: the paper is not a throwaway. Theorem 3 is a correct, citable result, and the failure mode of the theta-function approach is instructive. But the framing in the abstract and introduction is wrong, and the main claimed interpretation cannot be repaired within the current method. A serious referee should be sent this, with the expectation of major revision: keep Theorem 3 as the central contribution, state the symmetrized nature of the theta-function identities explicitly, and remove the claim that the full MacWilliams identity has been derived from theta functions.

Referee Report

3 major / 3 minor

Summary. The paper studies linear codes over the finite ring Z_k via lattices and theta functions. It formulates k auxiliary theta functions A_j(z), proves that the theta function of the lattice associated to a code C equals the complete weight enumerator of C evaluated at the A_j (Theorem 1), and then derives a functional identity for W_{C^⊥} evaluated at the same functions (Theorem 2). From this it obtains the symmetrized weight enumerator MacWilliams identity (Corollary 2). It further proves the genus-g complete weight enumerator MacWilliams identity for all positive k by Fourier transform and Poisson summation (Theorem 3), and for odd primes p gives an identity between the theta function of a lattice over a cyclotomic field and the complete weight enumerator of a code C ⊂ F_p^n with C ⊂ C^⊥ (Theorem 4). The paper's framing claims that these results extend the Bannai-Dougherty-Harada-Oura and van der Geer-Hirzebruch identities from even k to all k and from Lee to complete weight enumerators, respectively.

Significance. If all claims were valid, the paper would provide a theta-function interpretation of the MacWilliams identity for codes over Z_k for arbitrary k, a genus-g extension for all k, and a complete-weight-enumerator analogue of the Alpbach theorem. The paper contains some correct and useful pieces: Theorem 3 is a genuinely correct, self-contained proof of the genus-g complete weight enumerator MacWilliams identity for every positive integer k, extending the even-k case of Bannai et al.; Theorem 1 is a correct lattice-theta-function factorization; and the functional identity in Theorem 2 is correctly derived by Poisson summation. However, the central claimed implication from Theorem 2 to the polynomial complete weight enumerator MacWilliams identity fails because the required algebraic independence is not only unproved but is in fact false: the functions A_j satisfy A_j = A_{k-j}, so the theta map is never Zariski dense. Similarly, in the cyclotomic setting the functions ϑ_j satisfy ϑ_j = ϑ_{p-j}, so Theorem 4 does not deliver the advertised complete Hamming weight content.

major comments (3)

[Section III, Definition 7 and Eqs. (3.2)-(3.3)] For every j, A_j(z) = A_{k-j}(z) identically, because x ↦ -x is a bijection between kZ+j and kZ+(k-j) and the exponent in Definition 7 depends only on x^2. Hence the map z ↦ (A_0(z),...,A_{k-1}(z)) takes values in the proper closed subspace {X_j = X_{k-j}} of C^k, and for k ≥ 3 it is not Zariski dense in C^k. The authors state in Section VI that one must prove algebraic independence of A_0,...,A_{k-1} to show that Theorem 2 is equivalent to the complete weight enumerators MacWilliams identity; that algebraic independence is not merely unproved but impossible. Consequently the proof of Theorem 2 establishes only a functional identity on this symmetric subspace, and the only polynomial identity that can be concluded is the symmetrized identity of Corollary 2. The abstract's assertion that the complete weight enumerators MacWilliams identity is obtained from theta functions is therefore not supported by the proof.
[Section V, Theorem 4 and Eq. (5.1)] The same symmetry invalidates the advertised interpretation of Theorem 4. The ideal B is stable under negation, and Tr_{K^+/Q}(x x̄/p) is invariant under x ↦ -x, so ϑ_j(z) = ϑ_{p-j}(z) identically for every j. The identity ϑ_{Γ_C}(z) = W_C(ϑ_0(z),...,ϑ_{p-1}(z)) therefore reduces to the symmetrized Lee-type identity S_C(ϑ_0,...,ϑ_{(p-1)/2}), with the variables corresponding to the pairs {j,p-j} being redundant. Thus Theorem 4 is not a generalization of the van der Geer-Hirzebruch Alpbach theorem from Lee weight to complete Hamming weight; it is the Lee-weight identity written with duplicated variables. The same objection applies to the higher-dimensional version in Eq. (5.1).
[Section IV, Theorem 3] The proof of Theorem 3 by finite Fourier transform and Poisson summation is correct and is independent of the theta-function machinery. This theorem, giving the genus-g complete weight enumerator MacWilliams identity for all positive integers k, is the paper's principal sound result. However, because it is proved directly and does not use the A_j or ϑ_j functions, it does not rescue the paper's central claim of a theta-function derivation of the complete weight enumerator MacWilliams identity. A revised version would need to reframe the contribution around Theorem 3 and the symmetrized corollaries rather than around the invalid theta-function derivation.

minor comments (3)

[Abstract and Section VI] There are typographical errors: 'Hammi ng' in the abstract and 'fould' in the Conclusion should be 'Hamming' and 'found', respectively.
[Section IV, proof of Theorem 3] The notation w_a(y_j) in the Fourier transform computation is introduced rather abruptly; it would be clearer to define it explicitly as the indicator that the column y_j equals a, i.e., w_a(y_j) = 1 if y_j = a and 0 otherwise.
[Section V, proof of Theorem 4] In the factorization of the sum over ρ^{-1}(c), the notation ϑ_{c_i}(z) uses elements c_i ∈ F_p as indices; writing ϑ_{c_i mod p}(z) or ϑ_{[c_i]}(z) would avoid confusion with the integer representatives.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the MacWilliams identities are derived from Poisson summation and direct lattice factorizations rather than from the identities they assert.

full rationale

No circular step is present. Theorem 1 is obtained by decomposing rho^{-1}(c) into coordinate cosets, giving theta_{Gamma_C}(z)=prod_i A_{c_i}(z)=W_C(A_0,...,A_{k-1}); this is a factorization, not an assumption of the conclusion. Theorem 2 follows from Lemma 1 (Poisson summation for the one-variable A_j), Proposition 3, and Lemma 3; comparing (3.2) and (3.3) gives the functional identity, and the proof never invokes [47] or the desired polynomial MacWilliams identity. Theorem 3 is proved independently by Fourier transform on Z_k^g and Poisson summation, and Theorem 4 by the coset decomposition D=union(B+j). The self-citations [46] and [47] are contextual: [47] is mentioned for the already-known genus-1 identity and [46] for future nu-function work; neither carries a proof. I also flag Section VI's admission: 'To show that Theorem 2 is equivalent to the complete weight enumerators MacWilliams identity, it is important to prove the algebraic independence of A0,A1,...,Ak-1. We wish to find a proof...' This is a genuine missing-support/correctness gap, and the symmetry x -> -x gives A_j=A_{k-j}, so the sought independence cannot hold, but it is not a circular dependence: the functional identities are derived from first principles, not from their polynomial consequences.

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

The paper introduces k auxiliary theta functions A_j as mathematical tools, but these are new definitions rather than postulated physical entities. The main theorems rely on standard Fourier-analytic, lattice-theoretic, and algebraic number theory facts, plus one unproven algebraic independence property that the paper itself flags in its conclusion.

assumptions (5)

standard math Poisson summation formula for lattices and finite abelian groups
Invoked in Proposition 3, Lemma 1, and Theorem 3 to relate sums over a lattice or code to sums over its dual.
standard math Fourier transform of the Gaussian e^{-π i x^2/z} is (z/i)^{1/2} e^{π i z x^2}
Used in Lemma 1 and Proposition 3.
domain assumption Dual code satisfies |C||C⊥| = k^n and C⊥ annihilates C
Standard for finite Frobenius rings, cited to [10],[11] and used in Lemma 2.
standard math D/B ≅ F_p for B = (1-ξ) in the cyclotomic integers
Standard algebraic number theory; used in Section V to map codes over F_p to lattices in D^n.
ad hoc to paper Algebraic independence of A_0,...,A_{k-1} over C
Needed to promote the theta-functional identity of Theorem 2 to the ordinary polynomial MacWilliams identity; stated as unproven in Section VI.

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Pith. "Pith review of Codes over Finite Ring $\mathbb{Z}_k$, MacWilliams Identity and Theta Function." pith.science (2026). https://pith.science/paper/NNBTGNAY

@misc{pith2026250412604,
  author       = {Pith},
  title        = {Pith review of: Codes over Finite Ring $\mathbbZ_k$, MacWilliams Identity and Theta Function},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NNBTGNAY}},
  note         = {Machine review of arXiv:2504.12604}
}

abstract

In this paper, we study linear codes over $\mathbb{Z}_k$ based on lattices and theta functions. We obtain the complete weight enumerators MacWilliams identity and the symmetrized weight enumerators MacWilliams identity based on the theory of theta function. We extend the main work by Bannai, Dougherty, Harada and Oura to the finite ring $\mathbb{Z}_k$ for any positive integer $k$ and present the complete weight enumerators MacWilliams identity in genus $g$. When $k=p$ is a prime number, we establish the relationship between the theta function of associated lattices over a cyclotomic field and the complete weight enumerators with Hamming weight of codes, which is an analogy of the results by G. Van der Geer and F. Hirzebruch since they showed the identity with the Lee weight enumerators.

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