arXiv subjects

Wei-Hsuan Yu

Publications and source records attributed to Wei-Hsuan Yu.

36 records · Page 2Linked to original sources

arXiv2022

Semidefinite programming bounds for complex spherical codes

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.

math.CO↗

arXiv2022

Four-point semidefinite bound for equiangular lines

A set of lines in $\mathbb{R}^d$ passing through the origin is called equiangular if any two lines in the set form the same angle. We proved an alternative version of the three-point semidefinite constraints developed by Bachoc and Vallentin, and the multi-point semidefinite constraints developed by Musin for spherical codes. The alternative semidefinite constraints are simpler when the concerned object is a spherical $s$-distance set. Using the alternative four-point semidefinite constraints, we found the four-point semidefinite bound for equiangular lines. This result improves the upper bounds for infinitely many dimensions $d$ with prescribed angles. As a corollary of the bound, we proved the uniqueness of the maximum construction of equiangular lines in $\mathbb{R}^d$ for $7 \leq d \leq 14$ with inner product $α= 1/3$, and for $23 \leq d \leq 64$ with $α= 1/5$.

math.CO↗

arXiv2021

Improvement of generalization of Larman-Rogers-Seidel's theorem

A finite set $X$ in the $d$-dimensional Euclidean space is called an $s$-distance set if the set of distances between any two distinct points of $X$ has size $s$. In 1977, Larman-Rogers-Seidel proved that if the cardinality of an two-distance set is large enough, then there exists an integer $k$ such that the two distances $α$, $β$ $(α< β)$ having the integer condition, namely, $\frac{α^2}{β^2}=\frac{k-1}{k}$. In 2011, Nozaki generalized Larman-Rogers-Seidel's theorem to the case of $s$-distance sets, i.e. if the cardinality of an $s$-distance set $|X|\geqslant 2N$ with distances $α_1,α_2,\cdots,α_s$, where $N=\binom{d+s-1}{s-1}+\binom{d+s-2}{s-2}$, then the numbers $k_i=\prod_{j=1,2,\cdots,s,\text{ }j\neq i}\frac{α_{j}^{2}}{α_{j}^{2}-α_{i}^{2}}$ are integers. In this note, we reduce the lower bound of the requirement of integer condition of $s$-distance sets in $\mathbb{R}^d$. Furthermore, we can show that there are only finitely many $s$-distance sets $X$ in $\mathbb{R}^d$ with $|X|\geqslant 2\binom{d+s-1}{s-1}.$

math.CO↗

arXiv2020

Numerical computation of triangular complex spherical designs with small mesh ratio

This paper provides triangular spherical designs for the complex unit sphere $Ω^d$ by exploiting the natural correspondence between the complex unit sphere in $d$ dimensions and the real unit sphere in $2d-1$. The existence of triangular and square complex spherical $t$-designs with the optimal order number of points is established. A variational characterization of triangular complex designs is provided, with particular emphasis on numerical computation of efficient triangular complex designs with good geometric properties as measured by their mesh ratio. We give numerical examples of triangular spherical $t$-designs on complex unit spheres of dimension $d=2$ to $6$.

stat.ME↗

arXiv2020

Bounds on antipodal spherical designs with few angles

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 $X$ for these two cases is equivalent to estimating the size of real equiangular tight frames (ETFs) and Levenstein-equality packings, respectively. We first improve the previous estimate on the size of real ETFs and Levenstein-equality packings. This in turn gives a bound on $|X|$ when $s=3,\ t=3$ and $s=4,\ t=5$, respectively.

math.CO↗

arXiv2020

Semidefinite Programming Bounds For Spherical Three-distance Sets

A spherical three-distance set is a finite collection $X$ of unit vectors in $\mathbb{R}^{n}$ such that for each pair of distinct vectors has three inner product values. We use the semidefinite programming method to improve the upper bounds of spherical three-distance sets for several dimensions. We obtain better bounds in $\mathbb{R}^7$, $\mathbb{R}^{20}$, $\mathbb{R}^{21}$, $\mathbb{R}^{23}$, $\mathbb{R}^{24}$ and $\mathbb{R}^{25}$. In particular, we prove that maximum size of spherical three-distance sets is $2300$ in $\mathbb R^{23}$.

math.CO↗

arXiv2020

The Lemmens-Seidel conjecture and forbidden subgraphs

In this paper we show that the conjecture of Lemmens and Seidel of 1973 for systems of equiangular lines with common angle $\arccos (1/5)$ is true. Our main tool is forbidden subgraphs for smallest Seidel eigenvalue $-5$.

math.CO↗

arXiv2019

Equiangular lines and the Lemmens-Seidel conjecture

In this paper, claims by Lemmens and Seidel in 1973 about equiangular sets of lines with angle $1/5$ are proved by carefully analyzing pillar decompositions, with the aid of the uniqueness of two-graphs on $276$ vertices. The Neumann Theorem is generalized in the sense that if there are more than $2r-2$ equiangular lines in $\mathbb{R}^r$, then the angle is quite restricted. Together with techniques on finding saturated equiangular sets, we determine the maximum size of equiangular sets "exactly" in an $r$-dimensional Euclidean space for $r = 8$, $9$, and $10$.

math.CO↗

arXiv2018

Saturated configuration and new large construction of equiangular lines

A set of lines through the origin in Euclidean space is called equiangular when any pair of lines from the set intersects with each other at a common angle. We study the maximum size of equiangular lines in Euclidean space and use graph theoretic approach to prove that all the currently known construction for maximum equiangular lines in $\mathbb R^d$ cannot add another line to form a larger equiangular set of lines if $14 \leq d \leq 20$ and $d \neq 15$. We give new constructions of large equiangular lines which are 248 equiangular lines in $\mathbb R^{42}$, 200 equiangular lines in $\mathbb{R}^{41}$, 168 equiangular lines in $\mathbb{R}^{40}$, 152 equiangular lines in $\mathbb R^{39}$ with angle $1/7$, and 56 equiangular lines in $\mathbb R^{18}$ with angle $1/5$.

math.CO↗

arXiv2018

New code upper bounds for the folded n-cube

Let $Γ$ denote a distance-regular graph. The maximum size of codewords with minimum distance at least $d$ is denoted by $A(Γ,d)$. Let $\square_n$ denote the folded $n$-cube $H(n,2)$. We give an upper bound on $A(\square_n,d)$ based on block-diagonalizing the Terwilliger algebra of $\square_n$ and on semidefinite programming.The technique of this paper is an extension of the approach taken by A. Schrijver \cite{s} on the study of $A(H(n,2),d)$.

math.CO↗

arXiv2016

Upper bounds for $s$-distance sets and equiangular lines

The set of points in a metric space is called an $s$-distance set if pairwise distances between these points admit only $s$ distinct values. Two-distance spherical sets with the set of scalar products $\{α, -α\}$, $α\in[0,1)$, are called equiangular. The problem of determining the maximum size of $s$-distance sets in various spaces has a long history in mathematics. We suggest a new method of bounding the size of an $s$-distance set in compact two-point homogeneous spaces via zonal spherical functions. This method allows us to prove that the maximum size of a spherical two-distance set in $\mathbb{R}^n$, $n\geq 7$, is $\frac{n(n+1)}2$ with possible exceptions for some $n=(2k+1)^2-3$, $k \in \mathbb{N}$. We also prove the universal upper bound $\sim \frac 2 3 n a^2$ for equiangular sets with $α=\frac 1 a$ and, employing this bound, prove a new upper bound on the size of equiangular sets in all dimensions. Finally, we classify all equiangular sets reaching this new bound.

math.MG↗

arXiv2016

New bounds for equiangular lines and spherical two-distance sets

A set of lines in $\mathbb{R}^n$ is called equiangular if the angle between each pair of lines is the same. We derive new upper bounds on the cardinality of equiangular lines. Let us denote the maximum cardinality of equiangular lines in $\mathbb{R}^n$ with the common angle $\arccos α$ by $M_α(n)$. We prove that $M_{\frac 1 a} (n) \leq \frac 1 2 ( a^2-2) ( a^2-1)$ for any $n \in \mathbb{N}$ in the interval $a^2 -2 \leq n \leq 3 a^2-16$ and $a \geq 3$. Moreover, we discuss the relation between equiangular lines and spherical two-distance sets and we obtain the new results on the maximum spherical two-distance sets in $\mathbb{R}^n$ up to $n \leq 417$.

math.MG↗

arXiv2016

There are no 76 equiangular lines in $R^{19}$

Maximum size of equiangular lines in $\mathbb{R}^{19}$ has been known in the range between 72 to 76 since 1973. Acoording to the nonexistence of strongly regular graph $(75,32,10,16)$ \cite{aza15}, Larmen-Rogers-Seidel Theorem \cite{lar77} and Lemmen-Seidel bounds on equiangular lines with common angle $\frac 1 3$ \cite{lem73}, we can prove that there are no 76 equiangular lines in $\mathbb{R}^{19}$. As a corollary, there is no strongly regular graph $(76,35,18,14)$. Similar discussion can prove that there are no 96 equiangular lines in $\mathbb{R}^{20}$.

math.MG↗

arXiv2015

More on spherical designs of harmonic index $t$

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 paper we first study Fisher type lower bounds for the sizes of spherical designs of harmonic index $t$ (or for harmonic index $T$). We also study 'tight' spherical designs of harmonic index $t$ or index $T$. Here 'tight' means that the size of $Y$ attains the lower bound for this Fisher type inequality. The classification problem of tight spherical designs of harmonic index $t$ was started by Bannai-Okuda-Tagami (2015), and the case $t = 4$ was completed by Okuda-Yu (2015+). In this paper we show the classification (non-existence) of tight spherical designs of harmonic index 6 and 8, as well as the asymptotic non-existence of tight spherical designs of harmonic index $2e$ for general $e\geq 3$. We also study the existence problem for tight spherical designs of harmonic index $T$ for some $T$, in particular, including index $T = \{8,4\}$. We use (i) the linear programming method by Delsarte, (ii) the detailed information on the locations of the zeros as well as the local minimum values of Gegenbauer polynomials, (iii) the generalization by Hiroshi Nozaki of the Larman-Rogers-Seidel theorem on $2$-distance sets to $s$-distance sets, (iv) the theory of elliptic diophantine equations, and (v) the semidefinite programming method of eliminating some $2$-angular line systems for small dimensions.

math.CO↗

arXiv2015

Finite two-distance tight frames

A finite collection of unit vectors $S \subset \mathbb{R}^n$ is called a spherical two-distance set if there are two numbers $a$ and $b$ such that the inner products of distinct vectors from $S$ are either $a$ or $b$. We prove that if $a\ne -b,$ then a two-distance set that forms a tight frame for $\mathbb{R}^n$ is a spherical embedding of a strongly regular graph, and every strongly regular graph gives rise to two-distance tight frames through standard spherical embeddings. Together with an earlier work by S. Waldron on the equiangular case ({\em Linear Alg. Appl.}, vol. 41, pp. 2228-2242, 2009) this completely characterizes two-distance tight frames. As an intermediate result, we obtain a classification of all two-distance 2-designs.\

math.FA↗

arXiv2014

Nonexistence of tight spherical design of harmonic index 4

We give a new upper bound of the cardinality of a set of equiangular lines in $\R^n$ with a fixed angle $θ$ for each $(n,θ)$ satisfying certain conditions. Our techniques are based on semi-definite programming methods for spherical codes introduced by Bachoc--Vallentin [J.Amer.Math.Soc.2008]. As a corollary to our bound, we show the nonexistence of spherical tight designs of harmonic index 4 on $S^{n-1}$ with $n \geq 3$.

math.MG↗

arXiv2014

New bounds for equiangular lines

A set of lines in $\mathbb{R}^n$ is called equiangular if the angle between each pair of lines is the same. We address the question of determining the maximum size of equiangular line sets in $\mathbb{R}^n$, using semidefinite programming to improve the upper bounds on this quantity. Improvements are obtained in dimensions $24 \leq n \leq 136$. In particular, we show that the maximum number of equiangular lines in $\mathbb{R}^n$ is $276$ for all $24 \leq n \leq 41$ and is 344 for $n=43.$ This provides a partial resolution of the conjecture set forth by Lemmens and Seidel (1973).

math.MG↗

arXiv2013

New bounds for spherical two-distance sets

A spherical two-distance set is a finite collection of unit vectors in $\reals^n$ such that the set of distances between any two distinct vectors has cardinality two. We use the semidefinite programming method to compute improved estimates of the maximum size of spherical two-distance sets. Exact answers are found for dimensions $n=23$ and $40\le n\le 93\; (n\ne 46,78)$ where previous results gave divergent bounds.

math.MG↗