On the strong non-rigidity of certain tight Euclidean designs

Bannaĭ, Eiichi; Suprijanto, Djoko

doi:10.1016/j.ejc.2006.07.002

Cited by 10 publications

(5 citation statements)

References 13 publications

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“…The antipodal Euclidean tight 3-and 5-designs in Theorems 1.5 and 1.6 are non-rigid, because we obtain a distinct antipodal Euclidean tight 3-or 5-designs by changing one of the radii r i of the spheres which support the given antipodal Euclidean tight 3-or 5-design and the corresponding weight w(x), x ∈ X i . By a recent result on non-rigid Euclidean designs obtained by Eiichi Bannai and Djoko Suprijanto [4], it seems that if a Euclidean tight 2e-design or an antipodal Euclidean tight (2e + 1)-design which is supported by more than [ e+ε S 2 ] + 1 spheres exists, then there may possibly exist infinitely many Euclidean tight 2e-designs or antipodal Euclidean tight (2e + 1)-designs respectively. Actually they showed some of the tight Euclidean designs are strongly non-rigid.…”

Section: Remarkmentioning

confidence: 98%

On antipodal Euclidean tight (2e + 1)-designs

Bannaĭ

2006

J Algebr Comb

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Neumaier and Seidel (1988) generalized the concept of spherical designs and defined Euclidean designs in R n . For an integer t, a finite subset X of R n given together with a weight function w is a Euclidean t-design ifis the set of all the concentric spheres centered at the origin that intersect with X , X i = X ∩ S i , and w : X → R >0 . (The case of X ⊂ S n−1 with w ≡ 1 on X corresponds to a spherical t-design.) In this paper we study antipodal Euclidean (2e + 1)-designs. We give some new examples of antipodal Euclidean tight 5-designs. We also give the classification of all antipodal Euclidean tight 3-designs, the classification of antipodal Euclidean tight 5-designs supported by 2 concentric spheres.

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Section: Remarkmentioning

confidence: 98%

On antipodal Euclidean tight (2e + 1)-designs

Bannaĭ

2006

J Algebr Comb

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“…Step 2: If there exist 3-class symmetric association schemes on N 1 and N 2 points simultaneously, then further check if X i carries a Q-polynomial scheme (by Corollary 2.4). If such a polynomial scheme exists, then -Calculate the weight function w by equation (7)  and 3  .…”

Section: Omentioning

confidence: 99%

“…Moreover, they also completely classified tight Euclidean 4designs with constant weight in n  , for 2 n  , supported by two concentric spheres. Recently, in a joint work with Bannai and Bannai [7], the author introduced a new concept of strong non-rigidity for Euclidean t-designs. By using this new concept we also disproved Delsarte-Neumaier-Seidel's conjecture by showing the existence of infinitely many tight Euclidean designs having certain parameters.…”

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confidence: 99%

On Tight Euclidean 6-Designs: An Experimental Result

Suprijanto

2011

itbj.sci.

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“…As for the detailed definition and the basic properties of Euclidean designs and examples of Euclidean designs please refer [28,20,1,2,5,6,7,9,10,15,16,26,27,31], etc. Here we only give the fact we need directly to prove our main theorems.…”

Section: Some Basic Facts On Euclidean T-designsmentioning

confidence: 99%

Euclidean designs and coherent configurations

Bannaĭ¹

2010

Combinatorics and Graphs

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The concept of spherical t-design, which is a finite subset of the unit sphere, was introduced by Delsarte-Goethals-Seidel (1977). The concept of Euclidean tdesign, which is a two step generalization of spherical design in the sense that it is a finite weighted subset of Euclidean space, by Neumaier-Seidel (1988). We first review these two concepts, as well as the concept of tight t-design, i.e., the one whose cardinality reaches the natural lower bound. We are interested in t-designs (spherical or Euclidean) which are either tight or close to tight. As is well known by Delsarte-Goethals-Seidel (1977), in the study of spherical t-designs and in particular of those which are either tight or close to tight, association schemes play important roles. The main purpose of this paper is to show that in the study of Euclidean t-designs and in particular of those which are either tight or close to tight, coherent configurations play important roles. Here, coherent configuration is a purely combinatorial concept defined by D. G. Higman, and is obtained by axiomatizing the properties of general, not necessarily transitive, permutation groups, in the same way as association scheme was obtained by axiomatizing the properties of transitive permutation groups. The main purpose of this paper is to prove that Euclidean t-designs satisfying certain conditions give the structure of coherent configurations. In particular, it is seen that a tight Euclidean t-design on two concentric spheres centered at the origin has the structure of coherent configuration. Moreover, as an application of this general theory, we discuss the current status of our research to try to classify Euclidean 4-designs (X, w) on two concentric spheres S = S 1 ∪ S 2 centered at the origin whose weight function is constant on each X ∩ S i (i = 1, 2) and the number of the inner products between the distinct two points in X ∩ S i and X ∩ S j is at most 2 for i, j = 1, 2. We describe all the parameters of the coherent configurations, in terms of the parameters of the Euclidean designs. The classification of such Euclidean 4-designs is not yet completed, but we have found two new families of feasible parameters of such Euclidean 4-designs and the associated coherent configurations. One family corresponds to Euclidean tight 4-designs on two concentric spheres and another family is obtained from non-tight Euclidean 4-designs (and is related to the spherical tight 4-designs of one dimension more).

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On the strong non-rigidity of certain tight Euclidean designs

Cited by 10 publications

References 13 publications

On antipodal Euclidean tight (2e + 1)-designs

On antipodal Euclidean tight (2e + 1)-designs

On Tight Euclidean 6-Designs: An Experimental Result

Euclidean designs and coherent configurations

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