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The Bruck–Ryser–Chowla theorem is a result on the combinatorics of block designs. It states that if a (''v'', ''b'', ''r'', ''k'', λ)-design exists with ''v = b'' (a symmetric block design), then: * if ''v'' is even, then ''k'' − λ is a square; * if ''v'' is odd, then the following Diophantine equation has a nontrivial solution: *: ''x''2 − (''k'' − λ)''y''2 − (−1)(v−1)/2 λ ''z''2 = 0. The theorem was proved in the case of projective planes in . It was extended to symmetric designs in . == Projective planes == In the special case of a symmetric design with λ = 1, that is, a projective plane, the theorem (which in this case is referred to as the Bruck–Ryser theorem) can be stated as follows: If a finite projective plane of order ''q'' exists and ''q'' is congruent to 1 or 2 (mod 4), then ''q'' must be the sum of two squares. Note that for a projective plane, the design parameters are ''v'' = ''b'' = ''q''2 + ''q'' + 1, ''r'' = ''k'' = ''q'' + 1, λ = 1. Thus, ''v'' is always odd in this case. The theorem, for example, rules out the existence of projective planes of orders 6 and 14 but allows the existence of planes of orders 10 and 12. Since a projective plane of order 10 has been shown not to exist using a combination of coding theory and large-scale computer search, the condition of the theorem is evidently not sufficient for the existence of a design. However, no stronger general non-existence criterion is known. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Bruck–Ryser–Chowla theorem」の詳細全文を読む スポンサード リンク
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