EQUILATERAL POINT SETS IN ELLIPTIC GEOMETRY

Abstract

This chapter highlights equilateral point sets in elliptic geometry. Elliptic space of r−1 dimensions Er−1 is obtained from r-dimensional vector space Rr with inner product (a, b). For 1 <k<r, any k-dimensional linear subspace Rk of Rr is called a (k−1)-dimensional elliptic subspace Ek−1. The query for equilateral point sets in elliptic geometry leads to the search for matrices B of order n and elements whose smallest eigenvalue has a high multiplicity. For n elliptic points A1, A2, …, An, carried by the unit vectors a1, …, an and spanning elliptic space Er−1, the Gram matrix is symmetric, semipositive definite, and of rank r. B-matrices of order n ≡ 2r that have only two distinct eigenvalues with equal multiplicities r are called C-matrices. In view of the existence of a Hadamard matrix of order 92, it is interesting to know whether Paley's construction may be reversed to obtain a C-matrix of order 46.