We now use the theory of finite fields to construct a special type of matrix of use for applications in statistics and coding theory.

A Hadamard matrix of dimension n is an n by n matrix H all of whose entries are 1 or -1 such that

H HT = n In,

where In is the n by n identity matrix. The number n is called its dimension.

Hadamard

For n = 2, an example is

1 1
1 -1
.

One can show that the dimension of a Hadamard matrix is either 1, 2 or 0 mod 4. It is an open problem whether for each such n a Hadamard matrix of dimension n exists.

By a multiplicative procedure, one can construct Hadamard matrices of dimension mn from Hadamard matrices of dimensions m and n: if A = (aij)ij and B = (nkl)kl are Hadamard matrices then their Kronecker product is the matrix

A o B = (ai,j bk,l)ik,jl

of dimension mn.

Here we show how to make Hadamard matrices of various dimensions using finite fields.

Kronecker

Theorem

The following constructive results for Hadamard matrices are available.

  1. If A and B are Hadamard matrices, then so is A°B.
  2. If q is a prime power with q = 3 mod 4, then there exists a Hadamard matrix of dimension q + 1.
  3. If q is a prime power with q = 1 mod 4, then there exists a Hadamard matrix of dimension 2(q + 1).

Consequently, Hadamard matrices of order n exist for quite a few dimensions (multiples of 4). For n = 92, none of the above construction methods works. A Hadamard matrix of dimension 96 is known to exist, but even in the range of multiples of 4 less than 1000, existence is far from settled.