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Uniqueness of Certain Spherical Codes

Published online by Cambridge University Press:  20 November 2018

Eiichi Bannai  and
N. J. A. Sloane
Eiichi Bannai
Affiliation:
Ohio State University, Columbus, Ohio
N. J. A. Sloane
Affiliation:
Bell Laboratories, Murray Hill, New Jersey
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In this paper we show that there is essentially only one way of arranging 240 (resp. 196560) nonoverlapping unit spheres in R8 (resp. R24) so that they all touch another unit sphere, and only one way of arranging 56 (resp. 4600) spheres in R8 (resp. R24) so that they all touch two further, touching spheres. The following tight spherical t-designs are unique: the 5-design in Ω7, the 7-designs in Ω8 and Ω23, and the 11-design in Ω24. It was shown in [20] that the maximum number of nonoverlapping unit spheres in R8 (resp. R24) that can touch another unit sphere is 240 (resp. 196560). Arrangements of spheres meeting these bounds can be obtained from the E8 and Leech lattices, respectively. The present paper shows that these are the only arrangements meeting these bounds.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 2 , 01 April 1981 , pp. 437 - 449
Copyright
Copyright © Canadian Mathematical Society 1981

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