On orthogonal matrices with constant diagonal

In connection with the problem of finding the best projections of k-dimensional spaces embedded in n-dimensional spaces Hermann Konig asked: Given mER and nEN, are there n X n matrices C={c,,), i, i = 1,... ,n, such that c,,= m for all i, │C'ii│=l for i ≠ i, and C2={m2+n-l)ln? Konig was especially interested in symmetric C, and we find some families of matrices, satisfying this condition. We also find some families of matrices satisfying the less restrictive condition CCT = (m2 + n -1)1".