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Tsuyoshi Miezaki

Publications and source records attributed to Tsuyoshi Miezaki.

At least 37 records · Page 2Linked to original sources
arXiv2023

Equivariant theory for codes and lattices I

In this paper, we present a generalization of Hayden's theorem [7, Theorem 4.2] for $G$-codes over finite Frobenius rings. A lattice theoretical form of this generalization is also given. Moreover, Astumi's MacWilliams identity [1, Theorem 1] is generalized in several ways for different weight enumerators of $G$-codes over finite Frobenius rings. Furthermore, we provide the Jacobi analogue of Astumi's MacWilliams identity for $G$-codes over finite Frobenius rings. Finally, we study the relation between $G$-codes and its corresponding $G$-lattices.

math.CO↗
arXiv2023

Galois points for a finite graph

This paper introduces the notion of a Galois point for a finite graph, using the theory of linear systems of divisors for graphs discovered by Baker and Norine. We present a new characterization of complete graphs in terms of Galois points.

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arXiv2023

Weighted Tutte-Grothendieck polynomials of graphs

In this paper, we introduce the concept of the weighted (harmonic) chromatic polynomials of graphs and discuss some of its properties. We also present the notion of the weighted (harmonic) Tutte--Grothendieck polynomials of graphs and give a generalization of the recipe theorem between the harmonic Tutte--Grothendieck polynomials graphs and the harmonic Tutte polynomials of matroids. Moreover, we give some constructions of the weighted (harmonic) Tutte--Grothendieck invariants for graphs and the weighted (harmonic) Tutte invariants for matroids. Finally, we give a remark on the categorification of the harmonic chromatic polynomials of graphs and harmonic Tutte polynomials of matroids.

math.CO↗
arXiv2023

On the cycle index and the weight enumerator II

In a previous paper, the second and third named author introduced the concept of the complete cycle index and discussed a relation with the complete weight enumerator in coding theory. In the present paper, we introduce the concept of the complete joint cycle index and the average complete joint cycle index, and discuss a relation with the complete joint weight enumerator and the average complete joint weight enumerator respectively in coding theory. Moreover, the notion of the average intersection numbers is given, and we discuss a relation with the average intersection numbers in coding theory.

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arXiv2023

A note on the Assmus--Mattson theorem for some binary codes II

Let $C$ be a four-weight binary code, which has all one vector. Furthermore, we assume that $C$ supports $t$-designs for all weights obtained from the Assmus--Mattson theorem. We previously showed that $t\leq 5$. In the present paper, we show an analogue of this result in the cases of five and six-weight codes.

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arXiv2022

Jacobi polynomials and design theory I

In this paper, we introduce the notion of Jacobi polynomials with multiple reference vectors of a code, and give the MacWilliams type identity for it. Moreover, we derive a formula to obtain the Jacobi polynomials using the Aronhold polarization operator. Finally, we describe some facts obtained from Type III and Type IV codes that interpret the relation between the Jacobi polynomials and designs.

math.CO↗
arXiv2022

Harmonic Tutte polynomials of matroids

In the present paper, we introduce the concept of harmonic Tutte polynomials of matroids and discuss some of their properties. In particular, we generalize Greene's theorem, thereby expressing harmonic weight enumerators of codes as evaluations of harmonic Tutte polynomials.

math.CO↗
arXiv2022

Weight enumerators, intersection enumerators and Jacobi polynomials II

In the present paper, we introduce the concepts of Jacobi polynomials and intersection enumerators of codes over $\mathbb{F}_q$ and $\mathbb{Z}_{k}$ for arbitrary genus $g$. We also discuss the interrelation among them. Finally, we give the MacWilliams type identities for Jacobi polynomials.

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arXiv2022

On the support $t$-designs of extremal Type III and IV codes

Let $C$ be an extremal Type III or IV code and $D_{w}$ be the support design of $C$ for a weight $w$. We introduce the two numbers $δ(C)$ and $s(C)$: $δ(C)$ is the largest integer $t$ such that, for all weights, $D_{w}$ is a $t$-design; $s(C)$ denotes the largest integer $t$ such that there exists a $w$ such that $D_{w}$ is a $t$-design. In the present paper, we consider the possible values of $δ(C)$ and $s(C)$.

math.CO↗
arXiv2022

A note on the Assmus--Mattson theorem for some binary codes

We previously proposed the first nontrivial examples of a code having support $t$-designs for all weights obtained from the Assmus-Mattson theorem and having support $t'$-designs for some weights with some $t'>t$. This suggests the possibility of generalizing the Assmus-Mattson theorem, which is very important in design and coding theory. In the present paper, we generalize this example as a strengthening of the Assmus-Mattson theorem along this direction. As a corollary, we provide a new characterization of the extended Golay code $\mathcal{G}_{24}$.

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arXiv2021

Variants of Jacobi polynomials in coding theory

In this paper, we introduce the notion of the complete joint Jacobi polynomial of two linear codes of length $n$ over $\mathbb{F}_q$ and $\mathbb{Z}_k$. We give the MacWilliams type identity for the complete joint Jacobi polynomials of codes. We also introduce the concepts of the average Jacobi polynomial and the average complete joint Jacobi polynomial over $\mathbb{F}_q$ and $\mathbb{Z}_k$. We give a representation of the average of the complete joint Jacobi polynomials of two linear codes of length $n$ over $\mathbb{F}_q$ and $\mathbb{Z}_k$ in terms of the compositions of $n$ and its distribution in the codes. Further we present a generalization of the representation for the average of the $(g+1)$-fold complete joint Jacobi polynomials of codes over $\mathbb{F}_{q}$ and $\mathbb{Z}_{k}$. Finally, we give the notion of the average Jacobi intersection number of two codes.

math.CO↗
arXiv2021

Average of complete joint weight enumerators and self-dual codes

In this paper, we give a representation of the average of complete joint weight enumerators of two linear codes of length $n$ over $\mathbb{F}_{q}$ and $\mathbb{Z}_{k}$ in terms of the compositions of $n$ and their distributions in the codes. We also obtain a generalization of the representation for the average of $g$-fold complete joint weight enumerators of codes over $\mathbb{F}_{q}$ and $\mathbb{Z}_{k}$. Finally, the average of intersection numbers of a pair of Type III (resp. Type IV) codes, and its second moment are found.

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arXiv2020

A note on Assmus--Mattson type theorems

In the present paper, we give Assmus--Mattson type theorems for codes and lattices. We show that a binary doubly even self-dual code of length 24m with minimum weight 4m provides a combinatorial 1-design and an even unimodular lattice of rank 24m with minimum norm 2m provides a spherical 3-design. We remark that some of such codes and lattices give t-designs for higher t. As a corollary, we give some restrictions on the weight enumerators of binary doubly even self-dual codes of length 24m with minimum weight 4m. Ternary and quaternary analogues are also given.

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arXiv2020

Tutte polynomial, complete invariant, and theta series

In this study, we present two results that relate Tutte polynomials. First, we provide new and complete polynomial invariants for graphs. We note that the number of variables of our polynomials is one. Second, let L_1 and L_2 be two non-isomorphic lattices. We state that L_1 and L_2 are theta series equivalent if those theta series are the same. The problem of identifying theta series equivalent lattices is discussed in Prof.~Conway's book The Sensual (Quadratic) Form with the title "Can You Hear the Shape of a Lattice?" In this study, we present a method to find theta series equivalent lattices using matroids and their Tutte polynomials.

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