By Even S.

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Suppose that n ≡ 1, 2 (mod 4), and there exists a prime p ≡ 3 (mod 4) such that the largest power of p that divides n is odd. Then a projective plane of order n does not exist. 22 can be applied are n = 6, 14, 21, 22, and 30. Hence, projective planes of these orders do not exist. We now turn to the situation of arbitrary λ, where we derive an easy-touse corollary of the Bruck-Ryser-Chowla Theorem. Before proceeding to our main result, we define the concept of a quadratic residue. Suppose that m ≥ 2 is an integer and a is any integer.

15. Suppose q ≥ 2 is a prime power and d ≥ 2 is an integer. Then there exists a qd , qd−1 , qd−1 −1 q−1 -BIBD. Furthermore, if d > 2, there is a qd −1 qd−1 −1 q(qd−2 −1) q−1 , q−1 , q−1 -BIBD. 15 has the same parameters as q copies of PGd−1 (q). 4 The Bruck-Ryser-Chowla Theorem We now look at two necessary existence conditions for symmetric BIBDs, which are known (together) as the “Bruck-Ryser-Chowla Theorem”. 16 (Bruck-Ryser-Chowla Theorem, v even). Suppose there exists a symmetric (v, k, λ)-BIBD with v even.

5 Let M be the incidence matrix of a (v, b, r, k, 1)-BIBD and define N = M T M. Denote N = (ni,j ). Prove that ni,j = k if i = j 0 or 1 if i = j. 6 Construct a regular pairwise balanced design on six points that contains exactly four blocks of size three. 15. 17. 9 (a) Prove that no (6, 3, 2)-BIBD can contain repeated blocks. (b) Prove that all (6, 3, 2)-BIBDs are isomorphic. 22. 11 Show that all (7, 3, 1)-BIBDs are isomorphic by the following method. ) (a) Without loss of generality, we can take the points to be {1, .

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Algorithmic combinatorics by Even S.


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