Tsuyoshi Miezaki
In a previous study, we presented a construction of spherical 3-designs. In the current study, using this construction, we present new optimal antipodal spherical codes in the space of spherical harmonics. Our construction is a generalization of Bondarenko's work.
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We investigate several antipodal spherical designs on whether we can choose half of the points, one from each antipodal pair, such that they are balanced at the origin. In particular, root systems of type A, D and E, minimal points of Leech lattice and the unique tight 7-design on $S^{22}$ are studied. We also study a half of an antipodal spherical design from the viewpoint of association schemes and spherical designs of harmonic index $T$.
We prove that if X is a spherical t-design and s-distance set with $t\geq 2s-3$, then X has the structure of Q-polynomial association scheme of class s. Also, we describe the parameters of the association scheme.
Antipodality of spherical designs with odd harmonic indices
本文研究了具有特定调和指数的球面设计的最小规模，并通过区间设计证明了其反向性。
Existence and nonexistence of spherical $5$-designs of minimal type
本文研究极小类型球面5-设计的存在性与性质，证明紧球面5-设计为极小类型的充要条件及相关ETF构造条件，给出其不存在的算术条件，并推导对极球面4-距离5-设计为极小类型的必要条件。
A finite subset $X$ on the unit sphere $\mathbb{S}^{d-1}$ is called an $s$-distance set with strength $t$ if its angle set $A(X):=\{\langle \mathbf{x},\mathbf{y}\rangle : \mathbf{x},\mathbf{y}\in X,\mathbf{x}\neq\mathbf{y} \}$ has size $s$, and $X$ is a spherical $t$-design but not a spherical $(t+1)$-design. In this paper, we consider to estimate the maximum size of such antipodal set for small $s$. First, we improve the known bound on $|X|$ for each even integer $s\in[\frac{t+5}{2}, t+1]$ when $t\geq 3$. We next focus on two special cases: $s=3,\ t=3$ and $s=4,\ t=5$. Estimating the size of …
Designs in compact symmetric spaces and applications of great antipodal sets
本文综述紧致对称空间中作为设计的大对径集，并介绍2-数和大对径集在拓扑与群论中的应用。
Delsarte-Goethals-Seidel showed that if $X$ is a spherical $t$-design with degree $s$ satisfying $t\geq 2s-2$, $X$ carries the structure of an association scheme. Also Bannai-Bannai showed that the same conclusion holds if $X$ is an antipodal spherical $t$-design with degree $s$ satisfying $t=2s-3$. As a generalization of these results, we prove that a union of spherical designs with a certain property carries the structure of a coherent configuration. We derive triple regularity of tight spherical $4,5,7$-designs, mutually unbiased bases, linked symmetric designs with certain parameters.
Mutually unbiased weighing matrices (MUWM) are closely related to an antipodal spherical code with 4 angles. In the present paper, we clarify the relationship between MUWM and the spherical sets, and give the complete solution about the maximum size of a set of MUWM of weight 4 for any order. Moreover we describe some natural generalization of a set of MUWM from the viewpoint of spherical codes, and determine several maximum sizes of the generalized sets. They include an affirmative answer of the problem of Best, Kharaghani, and Ramp.
How can $d+k$ vectors in $\mathbb{R}^d$ be arranged so that they are as close to orthogonal as possible? In particular, define $θ(d,k):=\min_X\max_{x\neq y\in X}|\langle x,y\rangle|$ where the minimum is taken over all collections of $d+k$ unit vectors $X\subseteq\mathbb{R}^d$. In this paper, we focus on the case where $k$ is fixed and $d\to\infty$. In establishing bounds on $θ(d,k)$, we find an intimate connection to the existence of systems of ${k+1\choose 2}$ equiangular lines in $\mathbb{R}^k$. Using this connection, we are able to pin down $θ(d,k)$ whenever $k\in\{1,2,3,7,23\}$ and establ…
Spherical Designs with Infinite Harmonic Strength
研究d维球面上T为无限子集时的球面T-设计存在性问题，证明d≥2时有限子集具有无限谐波强度当且仅当其为对映集，并刻画了d=1时无限强度球面设计恰好是分圆设计。
The kissing number in 48 dimensions for codes with certain forbidden distances is 52 416 000
本文利用改进的 Delsarte–Goethals–Seidel 线性规划界，确定含特定禁用内积的 48 维对跖球面码接触数为 52 416 000，并证明相应同基数球面 11-设计极小。
Spherical designs and modular forms of the $D_4$ lattice
本文证明$D_4$格壳层构成对跖球面$\{10,4,2\}$-设计、2-壳层为紧设计且24点情形唯一，并将其应用于格的唯一性与模形式Fourier系数。
Spherical $t$-design is a finite subset on sphere such that, for any polynomial of degree at most $t$, the average value of the integral on sphere can be replaced by the average value at the finite subset. It is well-known that an equivalent condition of spherical design is given in terms of harmonic polynomials. In this paper, we define a spherical design of harmonic index $t$ from the viewpoint of this equivalent condition, and we give its construction and a Fisher type lower bound on the cardinality. Also we investigate whether there is a spherical design of harmonic index attaining the bou…
A finite subset $Y$ on the unit sphere $S^{n-1} \subseteq \mathbb{R}^n$ is called a spherical design of harmonic index $t$, if the following condition is satisfied: $\sum_{\mathbf{x}\in Y}f(\mathbf{x})=0$ for all real homogeneous harmonic polynomials $f(x_1,\ldots,x_n)$ of degree $t$. Also, for a subset $T$ of $\mathbb{N} = \{1,2,\cdots \}$, a finite subset $Y\subset S^{n-1}$ is called a spherical design of harmonic index $T,$ if $\sum_{\mathbf{x}\in Y}f(\mathbf{x})=0$ is satisfied for all real homogeneous harmonic polynomials $f(x_1,\ldots,x_n)$ of degree $k$ with $k\in T$. In the present pap…
Energy of codes with forbidden distances in 48 dimensions
该研究针对48维带禁用距离的编码能量问题，证明了4个球面11-设计分别在两类内积/距离禁用集合约束下的多类编码集合中具有普适最优性。
We find the set of all universal minimum points of the potential of the $16$-point sharp code on $S^4$ and (more generally) of the demihypercube on $S^d$, $d\geq 5$, as well as of the $2_{41}$ polytope on $S^7$. We also extend known results on universal minima of three sharp configurations on $S^{20}$ and $S^{21}$ containing no antipodal pair to their symmetrizations about the origin. Finally, we prove certain general properties of spherical $(2m-1)$-designs contained in as few as $m$ parallel hyperplanes (all but one configuration considered here possess this property).
The wave functions of a quantum isotropic harmonic oscillator in N-space modified by barriers at the coordinate hyperplanes can be expressed in terms of certain generalized spherical harmonics. These are associated with a product-type weight function on the sphere. Their analysis is carried out by means of differential-difference operators. The symmetries of this system involve the Weyl group of type B, generated by permutations and changes of sign of the coordinates. A new basis for symmetric functions as well as an explicit transition matrix to the monomial basis is constructed. This basis l…
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