Tight spherical designs, I

McLaren’s improved snub cube and other new spherical designs in three dimensions

“…which were established in [10], [3], [4]. In [16] we showed that 4-designs exist for N = 12, 14, ≥ 16, and conjectured that no others exist.…”

“…The regular snub cube ( [7], [9]), the familiar Archimedean solid with equal edges, has symmetry group [3,4] + . We take this group to consist of all even permutations of the three …”

Uniqueness of Certain Spherical Codes

“…The present paper shows that these are the only arrangements meeting these bounds. In [2], [3], it was shown that there are no tight spherical tdesigns for t ^ 8 except for the tight 11-design in fi 2 4-The present paper shows that this and three other tight /-designs are also unique. There is already a considerable body of literature concerning the uniqueness of these lattices and their associated codes and groups ( [5], [6], [8], [11], [13], [17]- [19], [21], [22], [27], [28]).…”

“…R 24 ) so that they all touch two further, touching spheres. The following tight spherical ^-designs are unique: the 5-design in fi 7 , the 7-designs in fi 8 and 12 2 3, and the 11-design in 12 2 4. It was shown in [20] that the maximum number of nonoverlapping unit spheres in R 8 (resp.…”

Self-dual codes over the Kleinian four group

“…The 196560 vectors of squared length 4 in the Leech lattice form the only tight spherical 11-design [BD79,BD80] in dimension greater then 2. This leads to the question: Is there a good notion of tight generalized t-designs, using a bound generalizing Theorem 20 for its definition, characterizing one of the two designs belonging to the Hexacode?…”