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| In [[mathematics]], the '''Paley construction''' is a method for constructing [[Hadamard matrix|Hadamard matrices]] using [[finite field]]s. The construction was described in 1933 by the [[England|English]] [[mathematician]] [[Raymond Paley]].
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| The Paley construction uses [[quadratic residue]]s 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).
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| ==Quadratic character and Jacobsthal matrix==
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| The quadratic [[character (mathematics)|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''<sup>2</sup> 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 = 1<sup>2</sup> = 6<sup>2</sup>, 4 = 2<sup>2</sup> = 5<sup>2</sup>, and 2 = 3<sup>2</sup> = 4<sup>2</sup>. Hence χ(0) = 0, χ(1) = χ(2) = χ(4) = 1, and χ(3) = χ(5) = χ(6) = −1.
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| 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
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| :<math>
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| Q = \begin{bmatrix}
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| 0 & -1 & -1 & 1 & -1 & 1 & 1\\
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| 1 & 0 & -1 & -1 & 1 & -1 & 1\\
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| 1 & 1 & 0 & -1 & -1 & 1 & -1\\
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| -1 & 1 & 1 & 0 & -1 & -1 & 1\\
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| 1 & -1 & 1 & 1 & 0 & -1 & -1\\
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| -1 & 1 & -1 & 1 & 1 & 0 & -1\\
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| -1 & -1 & 1 & -1 & 1 & 1 & 0\end{bmatrix}.
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| </math>
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| The Jacobsthal matrix has the properties ''QQ''<sup>T</sup> = ''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'')
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| 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
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| [[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.
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| ==Paley construction I==
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| If ''q'' is congruent to 3 (mod 4) then
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| :<math>
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| H=I+\begin{bmatrix}
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| 0 & j^T\\
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| -j & Q\end{bmatrix}
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| </math>
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| is a Hadamard matrix of size ''q'' + 1. Here ''j'' is the all-1 column vector of length ''q'' and ''I'' is the (''q''+1)×(''q''+1) identity matrix. The matrix ''H'' is a [[Hadamard matrix#Skew Hadamard matrices|skew Hadamard matrix]], which means it satisfies ''H''+''H''<sup>T</sup> = 2''I''.
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| ==Paley construction II==
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| If ''q'' is congruent to 1 (mod 4) then the matrix obtained by replacing all 0 entries in
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| :<math>
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| \begin{bmatrix}
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| 0 & j^T\\
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| j & Q\end{bmatrix}
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| </math>
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| with the matrix
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| :<math>
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| \begin{bmatrix}
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| 1 & -1\\
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| -1 & -1\end{bmatrix}
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| </math>
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| and all entries ±1 with the matrix
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| :<math>
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| \pm\begin{bmatrix}
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| 1 & 1\\
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| 1 & -1\end{bmatrix}
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| </math>
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| is a Hadamard matrix of size 2(''q'' + 1). It is a symmetric Hadamard matrix.
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| ==Examples==
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| Applying Paley Construction I to the Jacobsthal matrix for ''GF''(7), one produces the 8×8 Hadamard matrix,
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| <pre>
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| 11111111
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| -1--1-11
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| -11--1-1
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| -111--1-
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| --111--1
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| -1-111--
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| --1-111-
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| ---1-111.
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| </pre>
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| For an example of the Paley II construction when ''q'' is a prime power rather than a prime number, consider ''GF''(9). This is an [[field extension|extension field]] of ''GF''(3) obtained
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| by adjoining a root of an [[irreducible polynomial|irreducible quadratic]]. Different irreducible quadratics produce equivalent fields. Choosing ''x''<sup>2</sup>+''x''−1 and letting ''a'' be a root of this polynomial, the nine elements of ''GF''(9) may be written 0, 1, −1, ''a'', ''a''+1, ''a''−1, −''a'', −''a''+1, −''a''−1. The non-zero squares are 1 = (±1)<sup>2</sup>, −''a''+1 = (±''a'')<sup>2</sup>, ''a''−1 = (±(''a''+1))<sup>2</sup>, and −1 = (±(''a''−1))<sup>2</sup>. The Jacobsthal matrix is
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| :<math> | |
| Q = \begin{bmatrix}
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| 0 & 1 & 1 & -1 & -1 & 1 & -1 & 1 & -1\\
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| 1 & 0 & 1 & 1 & -1 & -1 & -1 & -1 & 1\\
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| 1 & 1 & 0 & -1 & 1 & -1 & 1 & -1 & -1\\
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| -1 & 1 & -1 & 0 & 1 & 1 & -1 & -1 & 1\\
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| -1 & -1 & 1 & 1 & 0 & 1 & 1 & -1 & -1\\
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| 1 & -1 & -1 & 1 & 1 & 0 & -1 & 1 & -1\\
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| -1 & -1 & 1 & -1 & 1 & -1 & 0 & 1 & 1\\
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| 1 & -1 & -1 & -1 & -1 & 1 & 1 & 0 & 1\\
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| -1 & 1 & -1 & 1 & -1 & -1 & 1 & 1 & 0\end{bmatrix}.
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| </math>
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| It is a symmetric matrix consisting of nine 3×3 circulant blocks. Paley Construction II produces the symmetric 20×20 Hadamard matrix,
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| <pre>
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| 1- 111111 111111 111111
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| -- 1-1-1- 1-1-1- 1-1-1-
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| 11 1-1111 ----11 --11--
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| 1- --1-1- -1-11- -11--1
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| 11 111-11 11---- ----11
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| 1- 1---1- 1--1-1 -1-11-
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| 11 11111- --11-- 11----
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| 1- 1-1--- -11--1 1--1-1
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| 11 --11-- 1-1111 ----11
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| 1- -11--1 --1-1- -1-11-
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| 11 ----11 111-11 11----
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| 1- -1-11- 1---1- 1--1-1
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| 11 11---- 11111- --11--
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| 1- 1--1-1 1-1--- -11--1
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| 11 ----11 --11-- 1-1111
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| 1- -1-11- -11--1 --1-1-
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| 11 11---- ----11 111-11
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| 1- 1--1-1 -1-11- 1---1-
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| 11 --11-- 11---- 11111-
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| 1- -11--1 1--1-1 1-1---.
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| </pre>
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| ==The Hadamard conjecture==
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| The size of a Hadamard matrix must be 1, 2, or a multiple of 4. The [[Kronecker product]] of two Hadamard matrices of sizes ''m'' and ''n'' is an Hadamard matrix of size ''mn''. By forming Kronecker products of matrices from the Paley construction and the 2×2 matrix,
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| :<math>
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| H_2 = \begin{bmatrix}
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| 1 & 1 \\
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| 1 & -1 \end{bmatrix},
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| </math>
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| Hadamard matrices of every allowed size up to 100 except for 92 are produced. In his 1933 paper, Paley says “It seems probable that, whenever ''m'' is divisible by 4, it is possible to construct an [[orthogonal matrix]] of order ''m'' composed of ±1, but the general theorem has every appearance of difficulty.” This appears to be the first published statement of the [[Hadamard conjecture]]. A matrix of size 92 was eventually constructed by Baumert, [[Solomon W. Golomb|Golomb]], and [[Marshall Hall (mathematician)|Hall]], using a construction due to Williamson combined with a computer search. Currently, Hadamard matrices have been shown to exist for all <math>\scriptstyle m \,\equiv\, 0 \mod 4</math> for ''m'' < 668.
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| ==See also==
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| *[[Paley biplane]]
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| *[[Paley graph]]
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| ==References==
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| * {{cite journal
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| | last = Paley
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| | first = R.E.A.C.
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| | authorlink = Raymond Paley
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| | title = On orthogonal matrices
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| | journal = Journal of Mathematics and Physics
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| | volume = 12
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| | issue =
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| | pages = 311–320
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| | publisher =
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| | location =
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| | year = 1933
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| | url =
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| | doi =
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| | id =
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| | accessdate = }}
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| * {{cite journal | author=L. D. Baumert | coauthors=[[Solomon W. Golomb|S. W. Golomb]], [[Marshall Hall (mathematician)|M. Hall Jr]] | title=Discovery of an Hadamard matrix of order 92 | journal=Bull. Amer. Math. Soc. | volume=68 | year=1962 | pages=237–238 | doi=10.1090/S0002-9904-1962-10761-7 | issue=3 }}
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| * {{cite book | author=F.J. MacWilliams | authorlink=Jessie MacWilliams | coauthors=[[Neil Sloane|N.J.A. Sloane]] | title=The Theory of Error-Correcting Codes | publisher=North-Holland | year=1977 | isbn=0-444-85193-3 | pages=47, 56 }}
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| [[Category:Matrices]]
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The title of the writer is Jayson. I am an invoicing officer and I'll be promoted quickly. Some time ago she chose to live in Alaska and her mothers and fathers reside nearby. I am really fond of handwriting but I can't make it my profession truly.
Feel free to surf to my web blog ... free psychic reading; click this link,