Paley construction

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 modulo 4.

Quadratic character and Jacobsthal matrix
Let q be a power of an odd prime. In the finite field GF(q) the quadratic character χ(a) indicates whether the element a is zero, a non-zero square, or a non-square:
 * $$\chi(a) = \begin{cases} 0 & \text{if }a = 0\\

1 & \text{if }a = b^2\text{ for some non-zero }b \in \mathrm{GF}(q)\\ -1 & \text{if }a\text{ is not the square of any element in }\mathrm{GF}(q).\end{cases}$$ 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&thinsp;×&thinsp;q matrix with rows and columns indexed by elements of GF(q) 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
 * $$Q = \begin{bmatrix}

0 & -1 & -1 & 1 & -1 & 1 & 1\\ 1 & 0 & -1 & -1 & 1 & -1 & 1\\ 1 & 1 & 0 & -1 & -1 & 1 & -1\\ -1 & 1 & 1 & 0 & -1 & -1 & 1\\ 1 & -1 & 1 & 1 & 0 & -1 & -1\\ -1 & 1 & -1 & 1 & 1 & 0 & -1\\ -1 & -1 & 1 & -1 & 1 & 1 & 0\end{bmatrix}.$$

The Jacobsthal matrix has the properties QQT = qI − J and QJ = JQ = 0 where I is the q&thinsp;×&thinsp;q identity matrix and J is the q&thinsp;×&thinsp;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 and rows and columns are indexed by field elements in the usual 0, 1, 2, … order, Q is a circulant matrix. That is, each row is obtained from the row above by cyclic permutation.

Paley construction I
If q is congruent to 3 mod 4 then


 * $$H = I + \begin{bmatrix}

0 & j^T \\ -j & Q \end{bmatrix}$$ 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 skew Hadamard matrix, which means it satisfies H + HT = 2I.

Paley construction II
If q is congruent to 1 mod 4 then the matrix obtained by replacing all 0 entries in


 * $$\begin{bmatrix}

0 & j^T \\ j & Q \end{bmatrix}$$

with the matrix


 * $$\begin{bmatrix}

1 & -1 \\ -1 & -1 \end{bmatrix}$$

and all entries ±1 with the matrix


 * $$\pm\begin{bmatrix}

1 & 1 \\ 1 & -1 \end{bmatrix}$$

is a Hadamard matrix of size 2(q + 1). It is a symmetric Hadamard matrix.

Examples
Applying Paley Construction I to the Jacobsthal matrix for GF(7), one produces the 8&thinsp;×&thinsp;8 Hadamard matrix,

11111111 -1--1-11 -11--1-1 -111--1- --111--1 -1-111-- --1-111- ---1-111.

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 extension field of GF(3) obtained by adjoining a root of an irreducible quadratic. Different irreducible quadratics produce equivalent fields. Choosing x2+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)2, −a+1 = (±a)2, a−1 = (±(a+1))2, and −1 = (±(a−1))2. The Jacobsthal matrix is


 * $$Q = \begin{bmatrix}

0 & 1 & 1 & -1 & -1 & 1 & -1 & 1 & -1\\ 1 & 0 & 1 & 1 & -1 & -1 & -1 & -1 & 1\\ 1 & 1 & 0 & -1 & 1 & -1 & 1 & -1 & -1\\ -1 & 1 & -1 & 0 & 1 & 1 & -1 & -1 & 1\\ -1 & -1 & 1 & 1 & 0 & 1 & 1 & -1 & -1\\ 1 & -1 & -1 & 1 & 1 & 0 & -1 & 1 & -1\\ -1 & -1 & 1 & -1 & 1 & -1 & 0 & 1 & 1\\ 1 & -1 & -1 & -1 & -1 & 1 & 1 & 0 & 1\\ -1 & 1 & -1 & 1 & -1 & -1 & 1 & 1 & 0\end{bmatrix}.$$

It is a symmetric matrix consisting of nine 3&thinsp;×&thinsp;3 circulant blocks. Paley Construction II produces the symmetric 20&thinsp;×&thinsp;20 Hadamard matrix,

1- 111111 111111 111111 -- 1-1-1- 1-1-1- 1-1-1-

11 1-1111 11 --11-- 1- --1-1- -1-11- -11--1 11 111-11 11 11 1- 1---1- 1--1-1 -1-11- 11 11111- --11-- 11 1- 1-1--- -11--1 1--1-1

11 --11-- 1-1111 11 1- -11--1 --1-1- -1-11- 11 11 111-11 11 1- -1-11- 1---1- 1--1-1 11 11 11111- --11-- 1- 1--1-1 1-1--- -11--1

11 11 --11-- 1-1111 1- -1-11- -11--1 --1-1- 11 11 11 111-11 1- 1--1-1 -1-11- 1---1- 11 --11-- 11 11111- 1- -11--1 1--1-1 1-1---.

The Hadamard conjecture
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&thinsp;×&thinsp;2 matrix,


 * $$H_2 = \begin{bmatrix}

1 & 1 \\ 1 & -1 \end{bmatrix},$$

Hadamard matrices of every permissible 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, Golomb, and Hall, using a construction due to Williamson combined with a computer search. Currently, Hadamard matrices have been shown to exist for all $$m \,\equiv\, 0 \mod 4$$ for m < 668.