Bounds on <i>s</i>-Distance Sets with Strength <i>t</i>

Nozaki, Hirōshi; Suda, Sho

doi:10.1137/100807053

Cited by 3 publications

(12 citation statements)

References 24 publications

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“…If X is not a spherical {3}-design, then X is a spherical 2-distance 2-design but not a 3-design. Therefore, in the latter case, by [15] or [16], we have |X| ≤ n(n + 1)/2. Consider the multiset X := X ∪ (−X), then |X | ≤ n(n + 1).…”

Section: Remarks From Nozakimentioning

confidence: 96%

Classification of Spherical $2$-distance $\{4,2,1\}$-designs by Solving Diophantine Equations

Bannaĭ

Xiang

Yu³

et al. 2021

Taiwanese J. Math.

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In algebraic combinatorics, the first step of the classification of interesting objects is usually to find all their feasible parameters. The feasible parameters are often integral solutions of some complicated Diophantine equations, which cannot be solved by known methods. In this paper, we develop a method to solve such Diophantine equations in 3 variables. We demonstrate it by giving a classification of finite subsets that are spherical 2-distance sets and spherical {4, 2, 1}-designs at the same time.

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Section: Remarks From Nozakimentioning

confidence: 96%

Classification of Spherical $2$-distance $\{4,2,1\}$-designs by Solving Diophantine Equations

Bannaĭ

Xiang

Yu³

et al. 2021

Taiwanese J. Math.

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“…Motivated by the methods developed in [23], we present an upper bound for |X| which lowers the Delsarte-Goethals-Seidel bound when s ∈ [ t+5 2 , t + 1] is an even integer and t 3.…”

Section: The General Casementioning

confidence: 99%

“…To our knowledge, one only finds two pairs (d, n) for which ETFs exist and achieve the size : (6, 16) and (22,176). The known configuration of these two ETFs is a subset of ETFs with parameters (7,28) and (23,276), respectively (see [28,Remark 5.2] and [27,Page 271]).…”

Section: The Case: S = 3 T =mentioning

confidence: 99%

“…Estimating the size of these designs provides a necessary condition on their existence. See [1,23] and the references for the recent work.…”

Section: Introductionmentioning

confidence: 99%

“…To our knowledge, so far, one only finds two pairs (d, n) for which Levenstein-equality packings achieve the size d(d+2) 2 9 : (7, 63) and (22,1408). The corresponding packings come from a tight spherical 7-design in R 8 and R 23 , respectively [21,Page 619]. We next explain the link between tight spherical 7-designs and Levensteinequality packings.…”

mentioning

confidence: 99%

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Bounds on Antipodal Spherical Designs with Few Angles

Yu³

2021

Electron. J. Combin.

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A finite subset $X$ on the unit sphere $\mathbb{S}^d$ 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 $X$ for small $s$. Motivated by the method developed by Nozaki and Suda, for each even integer $s\in[\frac{t+5}{2}, t+1]$ with $t\geq 3$, we improve the best known upper bound of Delsarte, Goethals and Seidel. 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 improve the previous estimate on the size of real ETFs and Levenstein-equality packings. This in turn gives an upper bound on $|X|$ when $s=3,\ t=3$ and $s=4,\ t=5$, respectively.

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Bounds on antipodal spherical designs with few angles

Xu¹,

Xu²,

Yu³

2020

Preprint

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A finite subset X on the unit sphere S d−1 is called an s-distance set with strength t if its angle set A(X) := { x, y : x, y ∈ X, x = 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 X for small s. Motivated by the method developed by Nozaki and Suda [NS11], for each even integer s ∈ [ t+5 2 , t + 1] with t ≥ 3, we improve the best known upper bound of Delsarte, Goethals and Seidel [DGS77]. 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 improve the previous estimate on the size of real ETFs and Levenstein-equality packings. This in turn gives an upper bound on |X| when s = 3, t = 3 and s = 4, t = 5, respectively.

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Bounds on s-Distance Sets with Strength t

Cited by 3 publications

References 24 publications

Classification of Spherical $2$-distance $\{4,2,1\}$-designs by Solving Diophantine Equations

Classification of Spherical $2$-distance $\{4,2,1\}$-designs by Solving Diophantine Equations

Bounds on Antipodal Spherical Designs with Few Angles

Bounds on antipodal spherical designs with few angles

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