Hadamard matrices can be constructed using finite field GF() when and is odd. Pick a representation relatively prime to . Then by coloring white (where is the floor function) distinct equally spaced residues mod (, , , ...; , , , ...; etc.) in addition to 0, a Hadamard matrix is obtained if the powers of (mod ) run through . For example,
(1)
|
is of this form with and . Since , we are dealing with GF(11), so pick and compute its residues (mod 11), which are
(2)
| |||
(3)
| |||
(4)
| |||
(5)
| |||
(6)
| |||
(7)
| |||
(8)
| |||
(9)
| |||
(10)
| |||
(11)
| |||
(12)
|
Picking the first residues and adding 0 gives: 0, 1, 2, 4, 5, 8, which should then be colored in the matrix obtained by writing out the residues increasing to the left and up along the border (0 through , followed by ), then adding horizontal and vertical coordinates to get the residue to place in each square.
(13)
|
To construct , consider the representations . Only the first form can be used, with and . We therefore use GF(19), and color 9 residues plus 0 white.
Now consider a more complicated case. For , the only form having is the first, so use the GF() field. Take as the modulus the irreducible polynomial , written 1021. A four-digit number can always be written using only three digits, since and . Now look at the moduli starting with 10, where each digit is considered separately. Then
(14)
|
Taking the alternate terms gives white squares as 000, 001, 020, 021, 022, 100, 102, 110, 111, 120, 121, 202, 211, and 221.