Spherical codes and Borsukʹs conjecture

The approach of Kalai and Kahn towards counterexamples of Borsukʹs conjecture is generalized to spherical codes. This allows the construction of a finite set in R323 which cannot be partitioned into 561 sets of smaller diameter, thus improving upon the previous known examples. The construction is based on the subset of vectors of minimal length in the Leech lattice.