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 × 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 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 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 + H^T = 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 × 8 Hadamard matrix,

{\begin{bmatrix}1&1&1&1&1&1&1&1\\-1&1&-1&-1&1&-1&1&1\\-1&1&1&-1&-1&1&-1&1\\-1&1&1&1&-1&-1&1&-1\\-1&-1&1&1&1&-1&-1&1\\-1&1&-1&1&1&1&-1&-1\\-1&-1&1&-1&1&1&1&-1\\-1&-1&-1&1&-1&1&1&1\end{bmatrix}}.

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 × 3 circulant blocks. Paley Construction II produces the symmetric 20 × 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 × 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.

References

Paley, R.E.A.C. (1933). "On orthogonal matrices". Journal of Mathematics and Physics. 12: 311–320. doi:10.1002/sapm1933121311. Zbl 0007.10004.
L. D. Baumert; S. W. Golomb; M. Hall Jr (1962). "Discovery of an Hadamard matrix of order 92". Bull. Amer. Math. Soc. 68 (3): 237–238. doi:10.1090/S0002-9904-1962-10761-7.
F.J. MacWilliams; N.J.A. Sloane (1977). The Theory of Error-Correcting Codes. North-Holland. pp. 47, 56. ISBN 0-444-85193-3.