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A n-set of equi-isoclinic planes in R^r is a set of n planes spanning R^reach pair of which has the same non-zero angle arccos(sqrt(lambda)). We provethat for any odd integer k such that 2k=p^alpha+1, p odd prime, alphanon-negative integer the maximum number of equi-isoclinic planes with anglearccos(sqrt(1/(2k-2))) in R^(2k-1) is equal to 2k-1. The solution of thisgeometric problem is obtained by the construction of complex conferencematrices of order 2k-1.

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