Searcharxiv⌕ Search

arXiv subjects

Tsuyoshi Miezaki

Publications and source records attributed to Tsuyoshi Miezaki.

At least 19 recordsLinked to original sources
arXiv2026

A New Approach to Code Smoothing Bounds

Code smoothing is a phenomenon in which an error distribution makes a code statistically close to the uniform distribution over the ambient space. This closeness is measured by the total variation distance. Recently, Debris-Alazard et al.\ introduced a smoothing bound, which is an upper bound on this total variation distance. Although the smoothing bound evaluates how the error distribution smooths a code, this bound applies only to linear codes. In this paper, we generalize this bound to not only linear codes but also specific non-linear codes. While the smoothing bound in previous work was obtained by Fourier analysis over finite abelian groups, we derive this bound using a graph-theoretic approach. To derive the smoothing bound, we consider code smoothing as the mixing of random walks on a specific graph, and use the concept of equitable partitions, which is well-studied in graph theory.

cs.IT↗
arXiv2026

Average hitting times and recurrence structures II: Cartesian products of powers of cycles and regular graphs

In our previous work \cite{MiezakiTamura2026}, we clarified the second-order linear recurrence structures appearing in the average hitting times on the $k$-th power graph $C_N^k$ of the cycle graph. In this paper, for a connected $r$-regular graph $G$ on $m$ vertices, we investigate the average hitting times of the simple random walk on the Cartesian product graph $C_N^k \square G$. By using discrete Fourier analysis in the $C_N^k$ direction and the Laplacian spectral decomposition of $G$, we decompose the average hitting time into a component proportional to the average hitting time on $C_N^k$ and correction terms arising from the nonzero Laplacian eigenspaces of $G$. For each nonzero Laplacian eigenvalue, we introduce a Chebyshev-type polynomial, and when all of its roots are simple, we express the correction term as a finite Green-type sum. Furthermore, for two vertices having the same $G$-coordinate, we transform this expression into a second-order linear recurrence representation of the form $V_\ell V_{N-\ell}/V_N$. When $G$ is a walk-regular graph, the average hitting time between two vertices having the same $G$-coordinate depends only on the Laplacian eigenvalues of $G$ and their multiplicities. We also derive formulas for the number of spanning trees and the number of two-component spanning forests of $C_N^k \square G$, and give several explicit examples.

math.CO↗
arXiv2026

Average Hitting Times and Recurrence Structures I: Powers of Cycle Graphs

We investigate the average hitting times of simple random walks on the $k$-th power graph $C_N^k$ of the cycle graph $C_N$. First, we show that the average hitting times are characterized by a difference equation corresponding to the graph Laplacian. Next, by using the cyclic symmetry of $C_N^k$, we derive a spectral representation via Fourier analysis. Furthermore, by applying factorization and partial fraction decomposition of the corresponding difference operator, we obtain an explicit formula for the average hitting times consisting of a quadratic term and finitely many correction terms. These correction terms are described by second-order linear recurrence sequences associated with the characteristic polynomials, and can be regarded as natural generalizations of Fibonacci-type sequences. As a consequence, our formulas recover the known results for cycle graphs and squares of cycle graphs in a unified way. Moreover, from the formulas obtained for average hitting times, we derive explicit formulas for the effective resistances, the numbers of spanning trees, the numbers of two-component spanning forests, and the numbers of spanning trees of vertex-identified graphs. In particular, for the third power graph $C_N^3$ of the cycle graph, all of these quantities are written explicitly in terms of complex conjugate Fibonacci-type sequences. Our results clarify structural relations between random walk quantities and combinatorial quantities on cycle power graphs.

math.CO↗
arXiv2025

Fibonacci and Lucas numbers arising from two-component spanning forests of wheel graphs

In this paper, we present a constructive bijection between a conditioned spanning forest of the wheel graph $W_{n+1}$ and a spanning tree of the fan graph $F_n$. In addition, by applying the effective resistance formula obtained by Bapat and Gupta \cite{bapat-gupta}, we derive an explicit formula for the number of two-component spanning forests of $W_{n+1}$ in which two specified vertices $u$ and $v$ lie in distinct components. Based on this result, we obtain explicit formulas for the following three conditioned two-component spanning forests $F_{W_{n+1}}(v_1\mid v_2)$, $F_{W_{n+1}}(v_1\mid v_3)$, and $F_{W_{n+1}}(v_1\mid v_c)$. These formulas are $F_{W_{n+1}}(v_1\mid v_2)=2(f_{2n-1}-1)$, $F_{W_{n+1}}(v_1\mid v_3)=2(\ell_{2n-2}-3)$, $F_{W_{n+1}}(v_1\mid v_c)=f_{2n}$, where $f_i$ and $\ell_j$ denote the $i$-th Fibonacci number and $j$-th Lucas number, respectively. As these identities show, the enumerations naturally lead to formulas involving Fibonacci numbers and Lucas numbers. Taken together, these two approaches show a unified perspective. One is the constructive combinatorial bijection, and the other is the analytic method based on effective resistance. Together they provide a new integrated framework for studying the structure of spanning forests on $W_{n+1}$.

math.CO↗
arXiv2025

Higher and extended Jacobi polynomials for codes

In this paper, we introduce Jacobi polynomial generalizations of several classical invariants in coding theory over finite fields, specifically, the higher and extended weight enumerators, and we establish explicit correspondences between the resulting Jacobi polynomials. Moreover, we present the Jacobi analogue of MacWilliams identity for both higher and extended weight enumerators. We also present that the higher Jacobi polynomials for linear codes whose subcode supports form $t$-designs can be uniquely determined from the higher weight enumerators of the codes via polarization technique. Finally, we demonstrate how higher Jacobi polynomials can be computed from harmonic higher weight enumerators with the help of Hahn polynomials.

math.CO↗
arXiv2025

On Lattice Isomorphism Problems for Lattices from LCD Codes over Finite Rings

These days, post-quantum cryptography based on the lattice isomorphism problem has been proposed. Ducas-Gibbons introduced the hull attack, which solves the lattice isomorphism problem for lattices obtained by Construction A from an LCD code over a finite field. Using this attack, they showed that the lattice isomorphism problem for such lattices can be reduced to the lattice isomorphism problem with the trivial lattice $\mathbb{Z}^n$ and the graph isomorphism problem. While the previous work by Ducas-Gibbons only considered lattices constructed by a code over a \textit{finite field}, this paper considers lattices constructed by a code over a \textit{finite ring} $\mathbb{Z}/k\mathbb{Z}$, which is a more general case. In particular, when $k$ is odd, an odd prime power, or not divisible by $4$, we show that the lattice isomorphism problem can be reduced to the lattice isomorphism problem for $\mathbb{Z}^n$ and the graph isomorphism problem.

cs.IT↗
arXiv2025

Harmonic higher and extended weight enumerators

In this paper, we present the harmonic generalizations of well-known polynomials of codes over finite fields, namely the higher weight enumerators and the extended weight enumerators, and we derive the correspondences between these weight enumerators. Moreover, we present the harmonic generalization of Greene's Theorem for the higher (resp. extended) weight enumerators. As an application of this Greene's-type theorem, we provide the MacWilliams-type identity for harmonic higher weight enumerators of codes over finite fields. Finally, we use this new identity to give a new proof of the Assmus-Mattson Theorem for subcode supports of linear codes over finite fields using harmonic higher weight enumerators.

math.CO↗
arXiv2025

The Terwilliger algebra of digraphs I -- Hamming digraph $H^*(d,3)$

In the present paper, we define the Terwilliger algebra of digraphs. Then, we determine the irreducible modules of the Terwilliger algebra of a Hamming digraph $H^*(d,3)$. As is well known, the representation of the Terwilliger algebra of a binary Hamming graph $H(d,2)$ is closely related to that of the Lie algebra $\mathit{sl}_2(\mathbb{C})$. We show that in the case of $H^*(d,3)$, it is related to that of the Lie algebra $\mathit{sl}_3(\mathbb{C})$. We also identify the Terwilliger algebra of $H^*(d,3)$ as the $d$ symmetric tensor algebra of ${\rm Mat}_3(\mathbb{C})$.

math.CO↗
arXiv2025

The graph zeta functions with respect to the group matrix of a finite group

In this paper, we present formulas for the edge zeta function and the second weighted zeta function with respect to the group matrix of a finite abelian group $Γ$. Furthermore, we give another proof of Dedekind Theorem for the group determinant of $Γ$ by the decomposition formula for a matrix of a group covering of a digraph. Finally, we treat the weighted complexity of the complete graph with entries of the group matrix of $Γ$ as arc weights.

math.CO↗
arXiv2025

Jacobi polynomials, invariant rings, and generalized $t$-designs

In the present paper, we provide results that relate the Jacobi polynomials in genus $g$. We show that if a code is $t$-homogeneous that is, the codewords of the code for every given weight hold a $t$-design, then its Jacobi polynomial in genus $g$ with composition $T$ with $|T|\leq t$ can be obtained from its weight enumerator in genus~$g$ using the polarization operator. Using this fact, we investigate the invariant ring, which relates the homogeneous Jacobi polynomials of the binary codes in genus $g$. Specifically, the generators of the invariant ring appearing for $g=1$ are obtained. Moreover, we define the split Jacobi polynomials in genus~$g$ and obtain the MacWilliams type identity for it. A split generalization for higher genus cases of the relation between the Jacobi polynomials and weight enumerator of a $t$-homogeneous code also given.

math.CO↗
arXiv2025

Universal graph series, chromatic functions, and their index theory

In the present paper, we introduce the concept of universal graph series. We then present four invariants of graphs and discuss some of their properties. In particular, one of these invariants is a generalization of the chromatic symmetric function and a complete invariant for graphs.

math.CO↗
arXiv2024

Neighbors, neighbor graphs and invariant rings in coding theory

In the present paper, we discuss the class of Type III and Type IV codes from the perspectives of neighbors. Our investigation analogously extends the results originally presented by Dougherty [8] concerning the neighbor graph of binary self-dual codes. Moreover, as an application of neighbors in invariant theory, we show that the ring of the weight enumerators of Type II code $d_{n}^{+}$ and its neighbors in arbitrary genus is finitely generated. Finally, we obtain a minimal set of generators of this ring up to the space of degree 24 and genus 3.

math.CO↗
arXiv2024

The Tutte polynomials of genus $g$

In the paper [Proceedings of the Japan Academy, Ser. A Mathematical Sciences, 95(10) 111-113], the authors introduce the concept of the Tutte polynomials of genus $g$ and announce that each matroid $M$ can be reconstructed from its Tutte polynomial of genus $|\mathcal{B}(M)|$, where $\mathcal{B}(M)$ denotes the family of bases of $M$. In that paper, we also announced that, for all $g$, there exist inequivalent matroids that have the same Tutte polynomial of genus $g$. In this paper, we prove these theorems.

math.CO↗
arXiv2023

Harmonic Tutte polynomials of matroids II

In this work, we introduce the harmonic generalization of the $m$-tuple weight enumerators of codes over finite Frobenius rings. A harmonic version of the MacWilliams-type identity for $m$-tuple weight enumerators of codes over finite Frobenius ring is also given. Moreover, we define the demi-matroid analogue of well-known polynomials from matroid theory, namely Tutte polynomials and coboundary polynomials, and associate them with a harmonic function. We also prove the Greene-type identity relating these polynomials to the harmonic $m$-tuple weight enumerators of codes over finite Frobenius rings. As an application of this Greene-type identity, we provide a simple combinatorial proof of the MacWilliams-type identity for harmonic $m$-tuple weight enumerators over finite Frobenius rings. Finally, we provide the structure of the relative invariant spaces containing the harmonic $m$-tuple weight enumerators of self-dual codes over finite fields.

math.CO↗