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Combinatorica

, Volume 31, Issue 6, pp 725–737 | Cite as

Inside S-inner product sets and Euclidean designs

  • Hiroshi Nozaki
Article

Abstract

A finite set X in the Euclidean space is called an s-inner product set if the set of the usual inner products of any two distinct points in X has size s. First, we give a special upper bound for the cardinality of an s-inner product set on concentric spheres. The upper bound coincides with the known lower bound for the size of a Euclidean 2s-design. Secondly, we prove the non-existence of 2- or 3-inner product sets on two concentric spheres attaining the upper bound for any d>1. The efficient property needed to prove the upper bound for an s-inner product set gives the new concept, inside s-inner product sets. We characterize the most known tight Euclidean designs as inside s-inner product sets attaining the upper bound.

Mathematics Subject Classification (2000)

05B30 52C99 

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Copyright information

© János Bolyai Mathematical Society and Springer Verlag 2011

Authors and Affiliations

  1. 1.Graduate School of Information SciencesTohoku UniversityAoba-ku, SendaiJapan

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