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In mathematics, the Paley construction is a method for constructing Hadamard matrices using finite fields. The construction was described in 1933 by the English mathematician Raymond Paley. The Paley construction uses quadratic residues in a finite field ''GF''(''q'') where ''q'' is a power of an odd prime number. There are two versions of the construction depending on whether ''q'' is congruent to 1 or 3 (mod 4). ==Quadratic character and Jacobsthal matrix== The quadratic character χ(''a'') indicates whether the given finite field element ''a'' is a perfect square. Specifically, χ(0) = 0, χ(''a'') = 1 if ''a'' = ''b''2 for some non-zero finite field element ''b'', and χ(''a'') = −1 if ''a'' is not the square of any finite field element. For example, in ''GF''(7) the non-zero squares are 1 = 12 = 62, 4 = 22 = 52, and 2 = 32 = 42. Hence χ(0) = 0, χ(1) = χ(2) = χ(4) = 1, and χ(3) = χ(5) = χ(6) = −1. The Jacobsthal matrix ''Q'' for ''GF''(''q'') is the ''q''×''q'' matrix with rows and columns indexed by finite field elements such that the entry in row ''a'' and column ''b'' is χ(''a'' − ''b''). For example, in ''GF''(7), if the rows and columns of the Jacobsthal matrix are indexed by the field elements 0, 1, 2, 3, 4, 5, 6, then : The Jacobsthal matrix has the properties ''QQ''T = ''qI'' − ''J'' and ''QJ'' = ''JQ'' = 0 where ''I'' is the ''q''×''q'' identity matrix and ''J'' is the ''q''×''q'' all-1 matrix. If ''q'' is congruent to 1 (mod 4) then −1 is a square in ''GF''(''q'') which implies that ''Q'' is a symmetric matrix. If ''q'' is congruent to 3 (mod 4) then −1 is not a square, and ''Q'' is a skew-symmetric matrix. When ''q'' is a prime number, ''Q'' is a circulant matrix. That is, each row is obtained from the row above by cyclic permutation. 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「Paley construction」の詳細全文を読む スポンサード リンク
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