It is shown that (n being an integer) any non-trivial finite linear space with n2- 1 points, all of degree at most n+1, is embeddable in a finite projective plane of order n. This generalizes a theorem of Bose and Shrikhande and settles the unsolved case n = 6.

The permanent of an n ✕ n matrix A = (aij) is the matrix function

1

where the summation is over all permutations in the symmetric group, Sn . An n ✕ n matrix A is a circulant if there are scalars a1 …, an such that

2

where P is the n ✕ n permutation matrix corresponding to the cycle (12 … n) in Sn.

An orthogonal design of order n and type (s1, s2) on the commuting variables x1, x2 is a matrix of order n with entries from {0, ± x1, ± x2} whose row vectors are formally orthogonal.

This note uses cyclotomy to construct orthogonal designs and finds several infinite families of new designs.

George Szekeres was born in Budapest on 29th May, 1911 the second of three sons to wealthy Jewish parents. As a youth he was shy and retiring, but early it become clear that his gifts lay in the direction of science and mathematics. At high school George was greatly influenced by his teacher in mathematics and physics, K. (Charles) Novobátzky, who worked actively in the theory of relativity and was in 1945 to become a professor of theoretical physics at the University of Budapest. Small wonder that George's first great mathematical interest was relativity. The other major formative influence at high school was the journal ‘Koözeépiskolai Matematikai és Fizikai Lapok’. The names of problem solver were published with each solution and with the completion of the year's volume, photographs of the main contributors were reproduced.

It is shown that if q is a prime power then there exists a circulant weighing matrix of order q2 + q + 1 with q2 nonzero elements per row and column.

This result allows the bound N to be lowered in the theorem of Geramita and Wallis that “given a square integer k there exists an integer N dependent on k such that weighing matrices of weight k and order n and orthogonal designs (1, k) of order 2n exist for every n > N”.

Infinite families of amicable orthogonal designs are constructed with

(i) both of type (1, q) in order q + 1 when q ≡ 3 (mod 4) is a prime power,

(ii) both of type (1, q) in order 2(q + 1) where q ≡ 1 (mod 4) is a prime power or q + 1 is the order of a conference matrix,

(iii) both of type (2, 2q) in order 2(q + 1) when q ≡ 1 (mod 4) is a prime power or q + 1 is the order of a conference matrix.