Paley construction

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 = b² 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

Q = [ 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 ] .

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.

Paley construction I

If q is congruent to 3 (mod 4) then

H = I + [ 0 j T − j Q ]

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

[ 0 j T j Q ]

with the matrix

[ 1 − 1 − 1 − 1 ]

and all entries ±1 with the matrix

± [ 1 1 1 − 1 ]

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,

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 = [ 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 ] .

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 = [ 1 1 1 − 1 ] ,

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, 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 ≡ 0 mod 4 for m < 668.