arXiv subjects
Publications and source records attributed to Sho Suda.
It follows from Delsarte theory that the Witt $4$-$(11,5,1)$ design gives rise to a $Q$-polynomial association scheme $\mathcal{W}$ defined on the set of its blocks. In this note we show that $\mathcal{W}$ is unique, i.e., defined up to isomorphism by its parameters.
Let $n$ be the order of a (quaternary) Hadamard matrix. It is shown that the existence of a projective plane of order $n$ is equivalent to the existence of a balancedly multi-splittable (quaternary) Hadamard matrix of order $n^2$.
A complex spherical code is a finite subset on the unit sphere in $\mathbb{C}^d$. A fundamental problem on complex spherical codes is to find upper bounds for those with prescribed inner products. In this paper, we determine the irreducible decomposition under the action of the one-point stabilizer of the unitary group $U(d)$ on the polynomial ring $\mathbb{C}[z_1\ldots,z_d,\bar{z}_1,\ldots,\bar{z}_d]$ in order to obtain the semidefinite programming bounds for complex spherical codes.
A weighing matrix $W$ is quasi-balanced if $|W||W|^\top=|W|^\top|W|$ has at most two off-diagonal entries, where $|W|_{ij}=|W_{ij}|$. A quasi-balanced weighing matrix $W$ signs a strongly regular graph if $|W|$ coincides with its adjacency matrix. Among other things, signed strongly regular graphs and their equivalent association schemes are presented.
A weighing matrix $W$ of order $n=\frac{p^{m+1}-1}{p-1}$ and weight $p^m$ is constructed and shown that the rows of $W$ and $-W$ form optimal constant weight ternary codes of length $n$, weight $p^m$ and minimum distance $p^{m-1}(\frac{p+3}{2})$ for each odd prime power $p$ and integer $m\ge 1$ and thus $$A_3\left(\frac{p^{m+1}-1}{p-1},p^{m-1}\big(\frac{p+3}{2}\big),p^{m}\right)=2\big(\frac{p^{m+1}-1}{p-1}\big).$$
We introduce a graph attached to mutually orthogonal Sudoku Latin squares. The spectra of the graphs obtained from finite fields are explicitly determined. As a corollary, we then use the eigenvalues to distinguish non-isomorphic Sudoku Latin squares.
Balanced weighing matrices with parameters $$ \left(1+18\cdot\frac{9^{m+1}-1}{8},9^{m+1},4\cdot 9^m\right), $$ for each nonzero integer $m$ is constructed. This is the first infinite class not belonging to those with classical parameters. It is shown that any balanced weighing matrix is equivalent to a five-class association scheme.
A unified approach to the construction of weighing matrices and certain symmetric designs is presented. Assuming the weight $p$ in a weighing matrix $W(n,p)$ is a prime power, it is shown that there is a $$W\left(\frac{p^{m+1}-1}{p-1}(n-1)+1,p^{m+1}\right)$$ for each positive integer $m$. The case of $n=p+1$ reduces to the balanced weighing matrices with classical parameters $$W\left(\frac{p^{m+2}-1}{p-1},p^{m+1}\right).$$ The equivalence with certain classes of association schemes is discussed in details.
Coherent configurations are a generalization of association schemes. In this paper, we introduce the concept of $Q$-polynomial coherent configurations and study the relationship among intersection numbers, Krein numbers, and eigenmatrices. The examples of $Q$-polynomial coherent configurations are provided from Delsarte designs in $Q$-polynomial schemes and spherical designs.
We show various upper bounds for the order of a digraph (or a mixed graph) whose Hermitian adjacency matrix has an eigenspace of prescribed codimension. In particular, this generalizes the so-called absolute bound for (simple) graphs first shown by Delsarte, Goethals, and Seidel (1977) and extended by Bell and Rowlinson (2003). In doing so, we also adapt the Blokhuis' theory (1983) of harmonic analysis in real hyperbolic spaces to that in complex hyperbolic spaces.
The notion of disjoint weighing matrices is introduced as a generalization of orthogonal designs. A recursive construction along with a computer search lead to some infinite classes of disjoint weighing matrices, which in turn are shown to form commutative association schemes with 3 or 4 classes.
Divisible design digraphs are constructed from skew balanced generalized weighing matrices and generalized Hadamard matrices. Commutative and non-commutative association schemes are shown to be attached to the constructed divisible design digraphs.
Deza digraphs were introduced in 2003 by Zhang and Wang as directed graph version of Deza graphs, that also generalize the notion of directed strongly regular graphs. In this paper we give several new constructions of Deza digraphs. Further, we introduce twin and Siamese twin (directed) Deza graphs and construct several examples. Moreover, we classify directed Deza graphs with parameters $(n,k,b,a,t)$ having the property that $b=t$. Finally, we introduce a variation of directed Deza graphs and provide a construction from finite fields.
For prime powers $q$ and $q+\varepsilon$ where $\varepsilon\in\{1,2\}$, an affine resolvable design from $\mathbb{F}_q$ and Latin squares from $\mathbb{F}_{q+\varepsilon}$ yield a set of symmetric designs if $\varepsilon=2$ and a set of symmetric group divisible designs if $\varepsilon=1$. We show that these designs derive commutative association schemes, and determine their eigenmatrices.
We complete the classification of tight $4$-designs in Hamming association schemes $H(n,q)$, i.e., that of tight orthogonal arrays of strength $4$, which had been open since a result by Noda (1979). To do so, we construct an association scheme attached to a tight $4$-design in $H(n,q)$ and analyze its triple intersection numbers to conclude the non-existence in all open cases.
The linked systems of symmetric group divisible designs of type II is introduced, and several examples are obtained from affine resolvable designs and mutually UFS Latin squares. Furthermore, an equivalence between such symmetric group divisible designs and some association schemes with $5$-classes is provided.
Balancedly splittable Hadamard matrices are introduced and studied. A connection is made to the Hadamard diagonalizable strongly regular graphs, maximal equiangular lines set, and unbiased Hadamard matrices. Several construction methods are presented. As an application, commutative association schemes of 4, 5, and 6 classes are constructed.
Let $X$ be a finite set in a complex sphere of $d$ dimension. Let $D(X)$ be the set of usual inner products of two distinct vectors in $X$. A set $X$ is called a complex spherical $s$-code if the cardinality of $D(X)$ is $s$ and $D(X)$ contains an imaginary number. We would like to classify the largest possible $s$-codes for given dimension $d$. In this paper, we consider the problem for the case $s=3$. Roy and Suda (2014) gave a certain upper bound for the cardinalities of $3$-codes. A $3$-code $X$ is said to be tight if $X$ attains the bound. We show that there exists no tight $3$-code except for dimensions $1$, $2$. Moreover we make an algorithm to classify the largest $3$-codes by considering representations of oriented graphs. By this algorithm, the largest $3$-codes are classified for dimensions $1$, $2$, $3$ with a current computer.