Periodicity of weight enumerators for codes generated by an integral matrix

In the theory of error-correcting codes, the minimum weight and the weight enumerator play a crucial role in evaluating the error-correcting capacity. In this paper, by viewing the weight enumerator as a quasi-polynomial, we reduce the calculation of the minimum weight to that of a code over a smaller integer residue ring. We also give a transformation formula between the Tutte quasi-polynomial and the weight enumerator. Furthermore, we compute the number of maximum weight codewords for the codes related to the matroids $N_k$ and $Z_k$. This is equivalent to computing the characteristic quasi-polynomial of the hyperplane arrangements related to $N_k$ and $Z_k$.

💡 Research Summary

The paper investigates linear codes generated by an integral matrix G and studies the weight enumerator W_G(x,y;q) as a quasi‑polynomial in the modulus q. By applying the Smith normal form to G, the authors extract elementary divisors e₁,…,e_r for each column subset J⊆E and define the lcm‑period ρ₀ as the least common multiple of the last elementary divisors of all non‑empty submatrices. This ρ₀ coincides with the period of the characteristic quasi‑polynomial χ_quasi^G(q) introduced by Kamiya‑Takemura‑Terao and later shown to be minimal by Higashitani‑Tran‑Yoshinaga.

The central contribution is Theorem 1.1, which gives a precise description of the periodic behavior of the minimum Hamming weight d_q of the code C_G(q) over ℤ/qℤ. Three cases are proved:

1. If gcd(q, ρ₀)=1, then d_q equals a universal constant d_{m₀}, where m₀ is the smallest integer >1 coprime to ρ₀. Thus all coprime moduli share the same minimum distance.
2. For any m≥2 that is coprime to the largest elementary divisor e_r, the minimum distance is constant on the set {q | gcd(q, ρ₀)=m}; i.e., d_q = d_m for every such q.
3. Conversely, if d_q = d_m holds for all q with gcd(q, ρ₀)=m, then m cannot divide the smallest elementary divisor e₁.

These statements imply that the entire infinite family {d_q | q∈ℕ} is determined by a finite subset of values, namely the d_m for the divisors m of ρ₀ that satisfy the coprimality condition. For example, when ρ₀=6, knowledge of d₂, d₃, d₅, and d₆ suffices to recover d_q for every q. The theorem also clarifies necessary conditions for the minimum distance to be globally constant.

The authors then express the weight enumerator coefficients A_{G,i}(q) as explicit functions of q. Lemma 3.1 shows that the size of any intersection of hyperplanes H_J(q) in the arrangement 𝔄_q is |H_J(q)| = q^{k−r(J)} ∏{j=1}^{r(J)} gcd(q, e{j,J}), where r(J) is the rank of the submatrix G_J. Using inclusion–exclusion, each coefficient A_{G,i}(q) becomes a sum of terms of the form q^{k−r} ∏ gcd(q, e), which are polynomials on each residue class modulo ρ₀. Consequently, W_G(x,y;q) is a monic quasi‑polynomial with period ρ₀, mirroring the behavior of the characteristic quasi‑polynomial of the associated hyperplane arrangement.

A significant theoretical bridge is built by extending Greene’s theorem. The classical relation between the weight enumerator of a linear code over a field and the Tutte polynomial of its associated matroid is generalized to the integer‑matrix setting: the Tutte quasi‑polynomial T_M^{quasi}(q) and the weight enumerator W_G(q) are linked by a transformation formula that reduces to Greene’s identity when q is prime. This provides a unified framework for interpreting combinatorial invariants of matroids, hyperplane arrangements, and coding theory within a quasi‑polynomial context.

In the applied part of the paper, the authors focus on two families of matroids, N_k and Z_k. They compute the number of codewords whose entries are all non‑zero (i.e., maximum weight) for the codes derived from these matroids. By employing a complete orthogonal system of additive characters modulo q, they translate the counting problem into evaluating character sums. The resulting expressions coincide with the m‑constituents of the characteristic quasi‑polynomial χ_quasi^G(q), thereby establishing that the maximum‑weight codeword count equals the value of the characteristic quasi‑polynomial on the appropriate residue class. Explicit formulas for these counts are presented, and the corresponding hyperplane arrangements’ characteristic quasi‑polynomials are derived as corollaries.

Section 6 supplies counterexamples that demonstrate the sharpness of Theorem 1.1. Example 6.1 shows that the converse of part (2) fails when the coprimality condition is violated, while Example 6.7 illustrates that part (3)’s converse does not hold even when m does not divide e₁. These examples underline that the theorem’s hypotheses are both necessary and essentially optimal.

Overall, the paper contributes a novel perspective by treating weight enumerators as quasi‑polynomials, thereby reducing the computation of minimum distances to a finite set of modular cases. It also enriches the interplay between coding theory, matroid theory, and the theory of hyperplane arrangements through the new transformation formula and explicit calculations for the N_k and Z_k families. The results open avenues for more efficient distance calculations, modular reduction techniques in code design, and deeper combinatorial interpretations of coding invariants.