Infinite family of equi-isoclinic planes in Euclidean odd dimensional spaces and of complex symmetric conference matrices of odd orders.

A n -set of equi-isoclinic planes in R r is a set of n planes spanning R r each pair of which has the same non-zero angle arccos ⁡ λ . We prove that for any odd integer k ≥ 3 such that 2 k = p α + 1 , p an odd prime, α a positive integer the maximum number of equi-isoclinic planes with angle arccos ⁡ 1 2 k − 2 in R 2 k − 1 is equal to 2 k − 1 . It is shown that the solution of this geometric problem is obtained by the construction of complex symmetric conference matrices of order 2 k − 1 , and that all these constructions are performed by use of the Legendre symbol of the Galois field G F ( p α ) . [ABSTRACT FROM AUTHOR]