Equilateral point sets in elliptic geometry

This chapter highlights equilateral point sets in elliptic geometry. Elliptic space of r−1 dimensions E r−1 is obtained from r -dimensional vector space R r with inner product ( a , b ). For 1 , any k -dimensional linear subspace R k of R r is called a ( k−1 )-dimensional elliptic subspace E k−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 A 1 , A 2 , …, A n , carried by the unit vectors a 1 , …, a n and spanning elliptic space E r−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.