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Akihiro Munemasa

Publications and source records attributed to Akihiro Munemasa.

At least 37 records · Page 2Linked to original sources
arXiv2021

Maximality of Seidel matrices and switching roots of graphs

In this paper, we discuss maximality of Seidel matrices with a fixed largest eigenvalue. We present a classification of maximal Seidel matrices of largest eigenvalue $3$, which gives a classification of maximal equiangular lines in a Euclidean space with angle $\arccos1/3$. Motivated by the maximality of the exceptional root system $E_8$, we define strong maximality of a Seidel matrix, and show that every Seidel matrix achieving the absolute bound is strongly maximal.

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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

Bordered complex Hadamard matrices and strongly regular graphs

We consider bordered complex Hadamard matrices whose core is contained in the Bose-Mesner algebra of a strongly regular graph. Examples include a complex Hadamard matrix whose core is contained in the Bose-Mesner algebra of a conference graph due to J. Wallis, F. Szöllősi, and a family of Hadamard matrices given by Singh and Dubey. In this paper, we prove that there are no other bordered complex Hadamard matrices whose core is contained in the Bose-Mesner algebra of a strongly regular graph.

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arXiv2019

Krein parameters of fiber-commutative coherent configurations

For fiber-commutative coherent configurations, we show that Krein parameters can be defined essentially uniquely. As a consequence, the general Krein condition reduces to positive semidefiniteness of finitely many matrices determined by the parameters of a coherent configuration. We mention its implications in the coherent configuration defined by a generalized quadrangle. We also simplify the absolute bound using the matrices of Krein parameters.

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arXiv2019

New Strongly Regular Graphs from Finite Geometries via Switching

We show that the strongly regular graph on non-isotropic points of one type of the polar spaces of type $U(n, 2)$, $O(n, 3)$, $O(n, 5)$, $O^+(n, 3)$, and $O^-(n, 3)$ are not determined by its parameters for $n \geq 6$. We prove this by using a variation of Godsil-McKay switching recently described by Wang, Qiu, and Hu. This also results in a new, shorter proof of a previous result of the first author which showed that the collinearity graph of a polar space is not determined by its spectrum. The same switching gives a linear algebra explanation for the construction of a large number of non-isomorphic designs.

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arXiv2019

Complex Hadamard matrices attached to a 3-class nonsymmetric association scheme

In this paper we classify complex Hadamard matrices contained in the Bose-Mesner algebra of nonsymmetric 3-class association schemes. As a consequence of our classification, we have two infinite families and some small examples of complex Hadamard matrices contained in the Bose-Mesner algebra of a self-dual fission of a complete multipartite graph.

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arXiv2019

Bent Vectorial Functions, Codes and Designs

Bent functions, or equivalently, Hadamard difference sets in the elementary Abelian group $(\gf(2^{2m}), +)$, have been employed to construct symmetric and quasi-symmetric designs having the symmetric difference property. The main objective of this paper is to use bent vectorial functions for a construction of a two-parameter family of binary linear codes that do not satisfy the conditions of the Assmus-Mattson theorem, but nevertheless hold $2$-designs. A new coding-theoretic characterization of bent vectorial functions is presented.

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arXiv2019

$D$-Magic and Antimagic Labelings of Hypercubes

For a set of distances $D$, a graph $G$ of order $n$ is said to be $D-$magic if there exists a bijection $f:V\rightarrow \{1,2, \ldots, n\}$ and a constant $k$ such that for any vertex $x$, $\sum_{y\in N_D(x)} f(y) =k$, where $N_D(x)=\{y|d(y,x)=j, j\in D\}$. In this paper we shall find sets of distances $D$s, such that the hypercube is $D-$magic. We shall utilise well-known properties of (bipartite) distance-regular graphs to construct the $D-$magic labelings.

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arXiv2018

Complementary Ramsey numbers and Ramsey graphs

In this paper, we consider a variant of Ramsey numbers which we call complementary Ramsey numbers $\bar{R}(m,t,s)$. We first establish their connections to pairs of Ramsey $(s,t)$-graphs. Using the classification of Ramsey $(s,t)$-graphs for small $s,t$, we determine the complementary Ramsey numbers $\bar{R}(m,t,s)$ for $(s,t)=(4,4)$ and $(3,6)$.

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arXiv2017

Butson-type complex Hadamard matrices and association schemes on Galois rings of characteristic 4

We consider nonsymmetric hermitian complex Hadamard matrices belonging to the Bose-Mesner algebra of commutative nonsymmetric association schemes. First, we give a characterization of the eigenmatrix of a commutative nonsymmetric association scheme of class 3 whose Bose-Mesner algebra contains a nonsymmetric hermitian complex Hadamard matrix, and show that such a complex Hadamard matrix is necessarily a Butson-type complex Hadamard matrix whose entries are 4-th roots of unity. We also give nonsymmetric association schemes X of class 6 on Galois rings of characteristic 4, and classify hermitian complex Hadamard matrices belonging to the Bose-Mesner algebra of X. It is shown that such a matrix is again necessarily a Butson-type complex Hadamard matrix whose entries are 4-th roots of unity.

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arXiv2017

A matrix approach to the Yang multiplication theorem

In this paper, we use two-variable Laurent polynomials attached to matrices to encode properties of compositions of sequences. The Lagrange identity in the ring of Laurent polynomials is then used to give a short and transparent proof of a theorem about the Yang multiplication.

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arXiv2017

Some restrictions on weight enumerators of singly even self-dual codes II

In this note, we give some restrictions on the number of vectors of weight $d/2+1$ in the shadow of a singly even self-dual $[n,n/2,d]$ code. This eliminates some of the possible weight enumerators of singly even self-dual $[n,n/2,d]$ codes for $(n,d)=(62,12)$, $(72,14)$, $(82,16)$, $(90,16)$ and $(100,18)$.

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arXiv2017

On a lower bound for the Laplacian eigenvalues of a graph

If $μ_m$ and $d_m$ denote, respectively, the $m$-th largest Laplacian eigenvalue and the $m$-th largest vertex degree of a graph, then $μ_m \geqslant d_m-m+2$. This inequality was conjectured by Guo in 2007 and proved by Brouwer and Haemers in 2008. Brouwer and Haemers gave several examples of graphs achieving equality, but a complete characterisation was not given. In this paper we consider the problem of characterising graphs satisfying $μ_m = d_m-m+2$. In particular we give a full classification of graphs with $μ_m = d_m-m+2 \leqslant 1$.

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arXiv2017

On $s$-extremal singly even self-dual $[24k+8,12k+4,4k+2]$ codes

A relationship between $s$-extremal singly even self-dual $[24k+8,12k+4,4k+2]$ codes and extremal doubly even self-dual $[24k+8,12k+4,4k+4]$ codes with covering radius meeting the Delsarte bound, is established. As an example of the relationship, $s$-extremal singly even self-dual $[56,28,10]$ codes are constructed for the first time. In addition, we show that there is no extremal doubly even self-dual code of length $24k+8$ with covering radius meeting the Delsarte bound for $k \ge 137$. Similarly, we show that there is no extremal doubly even self-dual code of length $24k+16$ with covering radius meeting the Delsarte bound for $k \ge 148$.

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arXiv2017

Nonexistence of certain singly even self-dual codes with minimal shadow

It is known that there is no extremal singly even self-dual $[n,n/2,d]$ code with minimal shadow for $(n,d)=(24m+2,4m+4)$, $(24m+4,4m+4)$, $(24m+6,4m+4)$, $(24m+10,4m+4)$ and $(24m+22,4m+6)$. In this paper, we study singly even self-dual codes with minimal shadow having minimum weight $d-2$ for these $(n,d)$. For $n=24m+2$, $24m+4$ and $24m+10$, we show that the weight enumerator of a singly even self-dual $[n,n/2,4m+2]$ code with minimal shadow is uniquely determined and we also show that there is no singly even self-dual $[n,n/2,4m+2]$ code with minimal shadow for $m \ge 155$, $m \ge 156$ and $m \ge 160$, respectively. We demonstrate that the weight enumerator of a singly even self-dual code with minimal shadow is not uniquely determined for parameters $[24m+6,12m+3,4m+2]$ and $[24m+22,12m+11,4m+4]$.

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arXiv2016

Complex Hadamard matrices attached to even orthogonal schemes of class 4

A complex Hadamard matrix is a square matrix W with complex entries of absolute value 1 satisfying WW*=nI, where * stands for the Hermitian transpose and I is the identity matrix of order n. In this paper, we give constructions of complex Hadamard matrices in the Bose-Mesner algebra of a certain 4-class symmetric association scheme. Moreover, we determine the Nomura algebras to show that the resulting matrices are not decomposable into nontrivial generalized tensor products.

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