Peter Boyvalenkov, Navid Safaei
Inspired by a recently formulated conjecture by Bannai et al. we investigate spherical codes which admit exactly three different distances and are spherical 5-designs. Computing and analyzing distance distributions we provide new proof of the fact (due to Levenshtein) that such codes are maximal and rule out certain cases towards a proof of the conjecture.
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Spherical two-distance sets and graph eigenvalues
本文建立了球面二距离集与特定谱性质图的显式对应，并据此推导了球面二距离集最大大小的界。
We Use the method of linearly independent polynomials to derive an upper bound for the cardinality of a spherical s-distance set F where the sum of distinct inner products of any two elements from F is zero. Our result generalizes the well-known Gerzon's bound for the cardinality of an equiangular spherical set to a significantly broader class of spherical s-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 $\mathbb R^{23}$.
We investigate spherical 4-distance 7-designs by studying their distance distributions. We compute these distance distributions and use their product (an integer) to derive certain divisibility conditions relating the dimension $n$ and the cardinality $M$ of our designs. It follows that $n$ divides $12M$ and $n+1$ divides $4M^2$. This result provides a good base for computer experiments to support the folklore conjecture that the only spherical 4-distance 7-designs are the tight spherical 7-designs. We then proceed with a computer assisted proof of this conjecture in all dimensions $n \leq 100…
A finite set $X$ in the Euclidean unit sphere is called an $s$-distance set if the set of distances between any distinct two elements of $X$ has size $s$. We say that $t$ is the strength of $X$ if $X$ is a spherical $t$-design but not a spherical $(t+1)$-design. Delsarte-Goethals-Seidel gave an absolute bound for the cardinality of an $s$-distance set. The results of Neumaier and Cameron-Goethals-Seidel imply that if $X$ is a spherical 2-distance set with strength 2, then the known absolute bound for 2-distance sets is improved. This bound are also regarded as that for a strongly regular graph…
Spectral conditions for spherical two-distance sets
本文研究了球面双距离集的谱条件，通过图谱分析确定其存在空间及最低维表示。
We prove that the inner products of spherical $s$-distance $t$-designs with $t \geq 2s-2$ (Delsarte codes) and $s \geq 3$ are rational with the only exception being the icosahedron. In other formulations, we prove that all sharp configurations have rational inner products and all spherical codes which attain the Levenshtein bound, have rational inner products, except for the icosahedron.
Every graph G can be embedded in a Euclidean space as a two-distance set. The Euclidean representation number of G is the smallest dimension in which G is representable by such an embedding. We consider spherical and J-spherical representation numbers of G and give exact formulas for these numbers using multiplicities of polynomials that are defined by the Caley-Menger determinant. One of the main results of the paper are explicit formulas for the representation numbers of the join of graphs which are obtained from W. Kuperberg's type theorem for two-distance sets.
We give a construction for spherical 3-designs. This construction is a generalization of Bondarenko's work.
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。
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.
We characterize that the image of the embedding of the $Q$-polynomial association scheme into eigenspace by primitive idempotent $E_1$ is a spherical $t$-design in terms of the Krein numbers. And we show that the strengths of $P$- and $Q$-polynomial schemes as spherical designs are bounded by constant.
Real spherical designs and real and complex projective designs have been shown by Delsarte, Goethals, and Seidel to give rise to association schemes when the strength of the design is high compared to its degree as a code. In contrast, designs on the complex unit sphere remain relatively uninvestigated, despite their importance in numerous applications. In this paper we develop the notion of a complex spherical design and show how many such designs carry the structure of an association scheme. In contrast with the real spherical designs and the real and complex projective designs, these associ…
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 give an excess theorem for spherical 2-designs. This theorem is a dual version of the spectral excess theorem for graphs, which gives a characterization of distance-regular graphs, among regular graphs in terms of the eigenvalues and the excess. Here we give a characterization of Q-polynomial association schemes among spherical 2-designs.
We derive upper and lower bounds on the sum of distances of a spherical code of size $N$ in $n$ dimensions when $N\sim n^α, 0<α\le 2.$ The bounds are derived by specializing recent general, universal bounds on energy of spherical sets. We discuss asymptotic behavior of our bounds along with several examples of codes whose sum of distances closely follows the upper bound.
We find out for which $t$ shells of selfdual lattices and of theirs shadows are spherical $t$-designs. The method uses theta series of lattices, which are modular forms. We analyse fully cubic and Witt lattices, as well as all selfdual lattices of rank at most 24.
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.
We derive upper bounds for the potential energy of spherical designs of cardinality close to the Delsarte-Goethals-Seidel bound. These bounds are obtained by linear programming with the use of the Hermite interpolating polynomial of the potential function in suitable nodes. Numerical computations show that the results are quite close to certain lower energy bounds confirming that spherical designs are, in a sense, energy efficient. \end{abstract}
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。
A set of vectors all of which have a constant (non-zero) norm value in an Euclidean lattice is called a shell of the lattice. Venkov classified strongly perfect lattices of minimum 3 (Réseaux et "designs" sphérique, 2001), whose minimal shell is a spherical 5-design. This note considers the classification of integral lattices whose shells of norm 3 are 5-designs.
An integral lattice which is generated by some vectors of norm $q$ is called $q$-lattice. Classification of 3-lattices of dimension at most four is given by Mimura (On 3-lattice, 2006). As a expansion, we give a classification of 3-lattices of dimension at most seven. In addition, we consider the spherical designs from its shells.
We derive general linear programming bounds for spherical $(k,k)$-designs. This includes lower bounds for the minimum cardinality and lower and upper bounds for minimum and maximum energy, respectively. As applications we obtain a universal bound in sense of Levenshtein for the minimum possible cardinality of a $(k,k)$ design for fixed dimension and $k$ and corresponding optimality result. We also discuss examples and possibilities for attaining the universal bound.
A set S of unit vectors in n-dimensional Euclidean space is called spherical two-distance set, if there are two numbers a and b, and inner products of distinct vectors of S are either a or b. The largest cardinality g(n) of spherical two-distance sets is not exceed n(n+3)/2. This upper bound is known to be tight for n=2,6,22. The set of mid-points of the edges of a regular simplex gives the lower bound L(n)=n(n+1)/2 for g(n. In this paper using the so-called polynomial method it is proved that for nonnegative a+b the largest cardinality of S is not greater than L(n). For the case a+b<0 we prop…
On $T$-avoiding spherical codes and designs in $\mathbb{R}^{32}$
本文研究了$\mathbb{R}^{32}$中极值偶数单模格的最短向量作为$T$-避免最优解的性质，并探讨了其作为球面设计和码的应用。
We apply polynomial techniques (linear programming) to obtain lower and upper bounds on the covering radius of spherical designs as function of their dimension, strength, and cardinality. In terms of inner products we improve the lower bounds due to Fazekas and Levenshtein and propose new upper bounds. Our approach to the lower bounds involves certain signed measures whose corresponding series of orthogonal polynomials are positive definite up to a certain (appropriate) degree. Upper bounds are based on a geometric observation and more or less standard linear programming techniques.
We extend the concepts of sum-free sets and Sidon-sets of combinatorial number theory with the aim to provide explicit constructions for spherical designs. We call a subset $S$ of the (additive) abelian group $G$ {\it $t$-free} if for all non-negative integers $k$ and $l$ with $k+l \leq t$, the sum of $k$ (not necessarily distinct) elements of $S$ does not equal the sum of $l$ (not necessarily distinct) elements of $S$ unless $k=l$ and the two sums contain the same terms. Here we shall give asymptotic bounds for the size of a largest $t$-free set in ${\bf Z}_n$, and for $t \leq 3$ discuss how …
We want to introduce a construction of spherical designs from finite graphs with the theory of crystal lattice. We start from a finite graph, and we consider standard realization of the crystal lattices as the maximal Abelian covering of the graph. Then, we take the set of vectors which form the crystal lattice. If every vector has the same norm, then we can consider a finite set on Euclidean sphere, and then we get a spherical design. In this paper, we observe the results by numerical calculations. We tried constructing vectors from various finite graphs, strongly regular graphs, distance reg…
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。
Spherical 2-Designs from Finite Group Orbits
本文通过同型分解和矩条件完整分类了由有限群轨道产生的球面2-设计，并证明其对应于正则表示中紧群轨道的投影，从而统一了几何与表示论视角。
A construction of spherical $5$-designs with $O(d^2)$ points
针对任意维度 $d$，利用基于 Sidon 集的复射影 $2$-设计构造，给出了一个至多 $72d^2$ 个点的等权球面 $5$-设计。
Evidence is presented to suggest that, in three dimensions, spherical 6-designs with N points exist for N=24, 26, >= 28; 7-designs for N=24, 30, 32, 34, >= 36; 8-designs for N=36, 40, 42, >= 44; 9-designs for N=48, 50, 52, >= 54; 10-designs for N=60, 62, >= 64; 11-designs for N=70, 72, >= 74; and 12-designs for N=84, >= 86. The existence of some of these designs is established analytically, while others are given by very accurate numerical coordinates. The 24-point 7-design was first found by McLaren in 1963, and -- although not identified as such by McLaren -- consists of the vertices of an "…
We show how the variational characterisation of spherical designs can be used to take a union of spherical designs to obtain a spherical design of higher order (degree, precision, exactness) with a small number of points. The examples that we consider involve taking the orbits of two vectors under the action of a complex reflection group to obtain a weighted spherical $(t,t)$-design. These designs have a high degree of symmetry (compared to the number of points), and many are the first known construction of such a design, e.g., a $32$ point $(9,9)$-design for $\mathbb{C}^2$, a $48$ point $(4,4…
Complex spherical designs from group orbits
本文研究有限群酉作用的轨道何时构成复球面设计，通过发展复设计理论与调和Molien级数，构造了若干推测最优的实和复球面t-设计。
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…
For each $N\ge C_dt^d$ we prove the existence of a well separated spherical $t$-design in the sphere $S^d$ consisting of $N$ points, where $C_d$ is a constant depending only on $d$.
On the existence and estimates of nested spherical designs
该研究证明了可通过在任意球面点集加额外点构造球面$t$-设计及嵌套球面设计的存在性，给出点数估计，证明特定条件下总点数上界阶为$t^{2d+1}$，并讨论了最优阶问题。
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。
Explicit construction of spherical $5$- and $7$-designs
本文提出通过提升紧融合框架的点集，显式构造球面5-和7-设计，并在不同维度下获得具有特定点数的球面设计。
Semidefinite programming bounds for distance distribution of spherical codes
本研究将球面码的半定规划界简化为单变量多项式不等式，通过四点论证得到六次多项式，从而简洁地证明了24胞腔在四维空间中 kissing 构型的唯一性，并给出了距离图限制。
Let X be a tight t-design of dimension n for one of the open cases t=5 or t=7. An investigation of the lattice generated by X using arithmetic theory of quadratic forms allows to exclude infinitely many values for n.
In this paper we investigate the Erdös/Falconer distance conjecture for a natural class of sets statistically, though not necessarily arithmetically, similar to a lattice. We prove a good upper bound for spherical means that have been classically used to study this problem. We conjecture that a majorant for the spherical means suffices to prove the distance conjecture(s) in this setting. For a class of non-Euclidean distances, we show that this generally cannot be achieved, at least in dimension two, by considering integer point distributions on convex curves and surfaces. In higher dimensions…
Let a \neq b be two positive scalars. A Euclidean representation of a simple graph G in R^r is a mapping of the nodes of G into points in R^r such that the squared Euclidean distance between any two points is a if the corresponding nodes are adjacent and b otherwise. A Euclidean representation is spherical if the points lie on an (r-1)-sphere, and is J-spherical if this sphere has radius 1 and a=2 < b. Let dim_E(G), dim_S(G) and dim_J(G) denote, respectively, the smallest dimension r for which G admits a Euclidean, spherical and J-spherical representation. In this paper, we extend and simplify…
Putatively optimal projective spherical designs with little apparent symmetry
该研究提出了几乎无对称性的推测最优投影球面设计的新显式实例与构建技术，通过数值研究明确了最优投影实球面设计的实代数簇的单点或无限族性质。
Spherical two-distance sets and eigenvalues of signed graphs
本研究聚焦高维固定双角度球面双距离集最大规模问题，提出极限值与符号图特征值重数相关的猜想，确定了两类参数下的极限值，是等角线外首个非平凡参数下的极限确定成果。
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。
Cohn--Conway--Elkies--Kumar [Experiment. Math. (2007)] described that one can construct a family of designs on $S^{2n-1}$ from a design on $\mathbb{CP}^{n-1}$. In this paper, we prove their claim for the case where $n=2$. That is, we give an algorithm to construct $2t$-designs on $S^{3}$ as products through a Hopf map $S^3 \rightarrow S^2$ of a $t$-design on $S^2$ and a $2t$-design on $S^1$.
Hybrid spherical designs
针对球面 $t$-设计曲线显式示例稀缺的问题，基于对偶凸多面体的顶点与边及其对称群多项式不变量，构造了新的球面 $t$-设计曲线，并提出结合点与曲线的混合 $t$-设计以实现更高次多项式精确积分，成功构造出 $t=19$ 的混合设计。
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…
Efficient lifts into odd dimensions of Spherical $5$- and $7$-designs with Balanced Halves
该研究提出将偶数维设计的平衡半集提升至合适纬度层的方法，构造奇数维球面5维和7维设计，得到最优二次阶的5维设计，假设Zauner猜想可改进界，还构造出特定点数的7维设计。
Let $μ$ be a Frostman measure on $E\subset\mathbb{R}^d$. The spherical average decay $$\int_{S^{d-1}}|\widehatμ(rω)|^2\,dω\lesssim r^{-β} $$ was originally used to attack Falconer distance conjecture, via Mattila's integral. In this paper we consider the pinned distance problem, a stronger version of Falconer distance problem, and show that spherical average decay implies the same dimensional threshold on both of them. In particular, with the best known spherical average estimates, we improve Peres-Schlag's result on pinned distance problem significantly. The idea is to reduce the pinned dista…
Energy bounds for weighted spherical codes and designs via linear programming
通过线性规划推导出加权球面码势能的通用界，证明在势函数严格绝对单调时最优性必要条件亦充分，并将结果扩展至加权球面设计。
In this article we investigate the $N$-point min-max and the max-min polarization problems on the sphere for a large class of potentials in $\mathbb{R}^n$. We derive universal lower and upper bounds on the polarization of spherical designs of fixed dimension, strength, and cardinality. The bounds are universal in the sense that they are a convex combination of potential function evaluations with nodes and weights independent of the class of potentials. As a consequence of our lower bounds, we obtain the Fazekas-Levenshtein bounds on the covering radius of spherical designs. Utilizing the exist…
Bounds on Discrete Potentials of Spherical (k,k)-Designs
本文推导了球面(k,k)-设计势极化问题的通用上下界，证明了其适用性，并展示了当势函数为$h(t)=t^{2k}$时该设计在相同基数球面码中的最优性。
We investigate universal bounds on spherical codes and spherical designs that could be obtained using Delsarte's linear programming methods. We give a lower estimate for the LP upper bound on codes, and an upper estimate for the LP lower bound on designs. Specifically, when the distance of the code is fixed and the dimension goes to infinity, the LP upper bound on codes is at least as large as the average of the best known upper and lower bounds. When the dimension n of the design is fixed, and the strength k goes to infinity, the LP bound on designs turns out, in conjunction with known lower …
More on the corner-vector construction for spherical designs
本文提出广义角向量方法构造加权球面设计，建立了一致度数上界，结合分析与组合技术证明，并给出度数7到13的多个示例。
We give an inequality on the packing of vectors/lines in quaternionic Hilbert space $\Hd$, which generalises those of Sidelnikov and Welch for unit vectors in $\Rd$ and $\Cd$. This has a parameter $t$, and depends only on the vectors up to projective unitary equivalence. The sequences of vectors in ${\mathbb{F}}^d={\mathbb{R}}^d,{\mathbb{C}}^d,{\mathbb{H}}^d$ that give equality, which we call spherical $(t,t)$-designs, are seen to satisfy a cubature rule on the unit sphere in ${\mathbb{F}}^d$ for a suitable polynomial space $\Hom_{\Fd}(t,t)$. Using this, we show that the projective spherical $…
Optimal asymptotic bounds for spherical designs
本文证明了Korevaar和Meyers的猜想：对于每个$N\ge c_d t^d$，存在由$N$个点组成的球面$t$-设计，其中$c_d$是仅依赖于$d$的常数。
Fixed-strength spherical designs
本文研究了固定t和维度趋于无穷大的情况下球面设计的最小大小，通过高斯设计构造更小的设计并确定了带符号t-设计的精确数量级。
We study manifolds $M$ equipped with a quadrature rule $$ \int_{M}{ϕ(x) dx} \simeq \sum_{i=1}^{n}{a_i ϕ(x_i)}.$$ We show that $n-$point quadrature rules with nonnegative weights on a compact $d-$dimensional manifold cannot integrate more than at most the first $c_{d}n + o(n)$ Laplacian eigenfunctions exactly. The constants $c_d$ are explicitly computed and $c_2 = 4$. The result is new even on $\mathbb{S}^2$ where it generalizes results on spherical designs.
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