Hadamard matrices H_n can be constructed using finite field GF(p^m) when p = 4l - 1 and m is odd. Pick a representation r relatively prime to p. Then by coloring white ⌊(p - 1)/2⌋ (where ⌊x⌋ is the floor function) distinct equally spaced residues mod p (r^0, r, r^2, ...; r^0, r^2, r^4, ...; etc.) in addition to 0, a Hadamard matrix is obtained if the powers of r (mod p) run through <⌊(p - 1)/2⌋. For example, n = 12 = 11^1 + 1 = 2(5 + 1) = 2^2(2 + 1) is of this form with p = 11 = 4×3 - 1 and m = 1.

Hadamard graph | Hadamard matrix | Paley class | Paley graph | Paley's theorem