Regular Two-Distance Sets

Casazza, Peter G.; Tran, Tin T.; Tremain, Janet C.

doi:10.1007/s00041-020-09756-4

Cited by 6 publications

(3 citation statements)

References 21 publications

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“…Many authors have studied finite sets of points over the unit sphere of R n having a small number of distinct distances among the points [1,12,16]; when this number is two, the sets are called spherical two-distance sets [2,3,4,8,11,24,25,29,30]. A two-distance tight frame is a real flat tight frame whose columns form a spherical two-distance set.…”

Section: Spherical Two-distance Sets and Two-distance Tight Framesmentioning

confidence: 99%

Constructions and Restrictions for Balanced Splittable Hadamard Matrices

Jedwab

Simon

2023

Electron. J. Combin.

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A Hadamard matrix is balanced splittable if some subset of its rows has the property that the dot product of every two distinct columns takes at most two values. This definition was introduced by Kharaghani and Suda in 2019, although equivalent formulations have been previously studied using different terminology. We collate previous results phrased in terms of balanced splittable Hadamard matrices, real flat equiangular tight frames, spherical two-distance sets, and two-distance tight frames. We use combinatorial analysis to restrict the parameters of a balanced splittable Hadamard matrix to lie in one of several classes, and obtain strong new constraints on their mutual relationships. An important consideration in determining these classes is whether the strongly regular graph associated with the balanced splittable Hadamard matrix is primitive or imprimitive. We construct new infinite families of balanced splittable Hadamard matrices in both the primitive and imprimitive cases. A rich source of examples is provided by packings of partial difference sets in elementary abelian $2$-groups, from which we construct Hadamard matrices admitting a row decomposition so that the balanced splittable property holds simultaneously with respect to every union of the submatrices of the decomposition.

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“…Many authors have studied finite sets of points over the unit sphere of R n having a small number of distinct distances among the points [1,12,16]; when this number is two, the sets are called spherical two-distance sets [2,3,4,8,11,24,25,29,30]. A two-distance tight frame is a real flat tight frame whose columns form a spherical two-distance set.…”

Section: Spherical Two-distance Sets and Two-distance Tight Framesmentioning

confidence: 99%

Constructions and Restrictions for Balanced Splittable Hadamard Matrices

Jedwab

Simon

2023

Electron. J. Combin.

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“…Many authors have studied finite sets of points over the unit sphere of R n having a small number of distinct distances among the points [1,12,16]; when this number is two, the sets are called spherical two-distance sets [2,3,4,8,11,23,24,28,29 (ii) A BSHM(n, ℓ, a, b) with ℓ > max{a, b} satisfies A two-distance tight frame is a real flat tight frame whose columns form a spherical two-distance set. A two-distance tight frame with values a, b where b = −a is a real flat ETF, as characterized in Result 2.14.…”

Section: 4mentioning

confidence: 99%

Constructions and restrictions for balanced splittable Hadamard matrices

Jedwab¹,

Li²,

Simon³

2022

Preprint

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A Hadamard matrix is balanced splittable if some subset of its rows has the property that the dot product of every two distinct columns takes at most two values. This definition was introduced by Kharaghani and Suda in 2019, although equivalent formulations have been previously studied using different terminology. We collate previous results phrased in terms of balanced splittable Hadamard matrices, real flat equiangular tight frames, spherical two-distance sets, and two-distance tight frames. We use combinatorial analysis to restrict the parameters of a balanced splittable Hadamard matrix to lie in one of several classes, and obtain strong new constraints on their mutual relationships. An important consideration in determining these classes is whether the strongly regular graph associated with the balanced splittable Hadamard matrix is primitive or imprimitive. We construct new infinite families of balanced splittable Hadamard matrices in both the primitive and imprimitive cases. A rich source of examples is provided by packings of partial difference sets in elementary abelian 2-groups, from which we construct Hadamard matrices admitting a row decomposition so that the balanced splittable property holds simultaneously with respect to every union of the submatrices of the decomposition.

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“…In the rank one case, the Geometric Brascamp-Lieb data is known as Parseval frame in coding theory and computer science (see for example Casazza, Tran, Tremain [33]).…”

Section: Introductionmentioning

confidence: 99%