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

TL;DR

This paper constructs an infinite family of equi-isoclinic planes in odd-dimensional Euclidean spaces by linking geometric configurations to complex conference matrices of odd orders, revealing new maximal arrangements.

Contribution

It establishes the maximum number of equi-isoclinic planes in certain odd-dimensional spaces and connects this to the construction of complex conference matrices of odd order.

Findings

Maximum number of equi-isoclinic planes in R^{2k-1} is 2k-1 for specified k.

Constructs complex conference matrices of order 2k-1.

Provides a geometric interpretation of complex conference matrices.

Abstract

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(sqrt(lambda)). We prove that for any odd integer k such that 2k=p^alpha+1, p odd prime, alpha non-negative integer the maximum number of equi-isoclinic planes with angle arccos(sqrt(1/(2k-2))) in R^(2k-1) is equal to 2k-1. The solution of this geometric problem is obtained by the construction of complex conference matrices of order 2k-1.