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Periodicity of weight enumerators for codes generated by an integral matrix

Koji Imamura, Norihiro Nakashima, Takuya Saito

TL;DR

This work analyzes the periodic structure of weight enumerators for codes generated by integer matrices by treating the weight distribution as a quasi-polynomial whose period is governed by the elementary divisors of the generator matrix. It develops a comprehensive framework—linking weight enumerators to characteristic and Tutte quasi-polynomials via a Greene-type identity, and expressing the weight distribution through constituents $f_i^m(t)$—to reduce minimum-weight computations over $\mathbb{Z}_q$ to smaller moduli. The authors provide explicit formulas for the constituents, establish gcd-based criteria for when minimum weights stabilize across residue classes, and compute exact characteristic quasi-polynomials for the Nk and Zk matroids, showing the minimum period equals $\mathrm{lcm}(1,2,\dots,k-1)$. They also include extensive examples and an appendix with computational data illustrating the phenomena and parity obstructions. Overall, the paper deepens the connection between coding theory, matroid theory, and hyperplane arrangements, offering practical tools to assess minimum distances and weight distributions through quasi-polynomial mechanisms.

Abstract

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 $Z_k$.

Periodicity of weight enumerators for codes generated by an integral matrix

TL;DR

This work analyzes the periodic structure of weight enumerators for codes generated by integer matrices by treating the weight distribution as a quasi-polynomial whose period is governed by the elementary divisors of the generator matrix. It develops a comprehensive framework—linking weight enumerators to characteristic and Tutte quasi-polynomials via a Greene-type identity, and expressing the weight distribution through constituents —to reduce minimum-weight computations over to smaller moduli. The authors provide explicit formulas for the constituents, establish gcd-based criteria for when minimum weights stabilize across residue classes, and compute exact characteristic quasi-polynomials for the Nk and Zk matroids, showing the minimum period equals . They also include extensive examples and an appendix with computational data illustrating the phenomena and parity obstructions. Overall, the paper deepens the connection between coding theory, matroid theory, and hyperplane arrangements, offering practical tools to assess minimum distances and weight distributions through quasi-polynomial mechanisms.

Abstract

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 and . This is equivalent to computing the characteristic quasi-polynomial of the hyperplane arrangements related to and .
Paper Structure (10 sections, 23 theorems, 94 equations, 3 figures, 3 tables)

This paper contains 10 sections, 23 theorems, 94 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries on weight enumerators and characteristic quasi-polynomials
  3. Weight enumerators
  4. Characteristic quasi-polynomials
  5. Quasi-polynomial of weight distribution
  6. Constituents of the weight enumerator
  7. Periods of minimum weights
  8. The Greene type identity
  9. Characteristic quasi-polynomial of certain classes of codes
  10. Appendix: sample computations

Key Result

Theorem 1.1

Let $G$ be a $k\times n$ matrix whose entries are integers, and let $\rho_0$ be the lcm period of $G$. Let $e_1|e_2|\cdots|e_{r}$ be the elementary divisors of $G$, where $r$ is the rank of $G$. Let $m\in\mathbb{Z}_{>0}$ be a divisor of $\rho_0$. Then the following hold.

Figures (3)

  • Figure 1: Hyperplanes and intersection points of the arrangement in Example \ref{['example-B2-20/02']}(1) when $q=5$
  • Figure 2: Hyperplanes and intersection points of the arrangement in Example \ref{['example-B2-20/02']}(1) when $q=6$
  • Figure 3: Hyperplanes and intersection points of the arrangement in Example \ref{['example-B2-20/02']}(2) when $q=4$

Theorems & Definitions (55)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Remark 2.5
  • Remark 2.6
  • Example 2.7
  • Theorem 2.8: Kamiya, Takemura, and Terao KTT08
  • Theorem 2.9: Higashitani, Tran, and Yoshinaga HTY2023
  • ...and 45 more