Uniqueness of Certain Spherical Codes

Abstract

In this paper we show that there is essentially only one way of arranging 240 (resp. 196560) nonoverlapping unit spheres in R 8 (resp. R 24 ) so that they all touch another unit sphere, and only one way of arranging 56 (resp. 4600) spheres in R 8 (resp. R 24 ) so that they all touch two further, touching spheres. The following tight spherical t -designs are unique: the 5-design in Ω 7 , the 7-designs in Ω 8 and Ω 23 , and the 11-design in Ω 24 . It was shown in [20] that the maximum number of nonoverlapping unit spheres in R 8 (resp. R 24 ) that can touch another unit sphere is 240 (resp. 196560). Arrangements of spheres meeting these bounds can be obtained from the E 8 and Leech lattices, respectively. The present paper shows that these are the only arrangements meeting these bounds.