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Inside S-inner product sets and Euclidean designs

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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.

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Authors and Affiliations

  1. Graduate School of Information Sciences, Tohoku University, Aramaki-Aza-Aoba 6-3-09, Aoba-ku, Sendai, 980-8579, Japan

    Hiroshi Nozaki

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  1. Hiroshi Nozaki
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Correspondence to Hiroshi Nozaki.

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Supported by JSPS Research Fellow.

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Nozaki, H. Inside S-inner product sets and Euclidean designs. Combinatorica 31, 725–737 (2011). https://doi.org/10.1007/s00493-011-2661-0

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  • DOI: https://doi.org/10.1007/s00493-011-2661-0

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