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 Rr is a set of n planes spanning Rr each pair of which has the same non-zero angle arccos⁡λ. We prove that for any odd integer k≥3 such that 2k=pα+1, p an odd prime, α a positive integer the maximum number of equi-isoclinic planes with angle arccos⁡12k−2 in R2k−1 is equal to 2k−1. It is shown that the solution of this geometric problem is obtained by the construction of complex symmetric conference matrices of order 2k−1, and that all these constructions are performed by use of the Legendre symbol of the Galois field GF(pα).