The Algebra of Two Dimensional Patterns

One way to create a random array is to map a random sequence into a two-dimensional pattern and an obvious choice is the use of shift-register sequences [10] or to use prime reciprocal sequences [11]-[14]. One dimensional binary random sequences are obtained as expansions of the prime reciprocal 1/p by [11],[12], a(i) = 2 mod p mod 2. Shift register (maximum length) sequences are obtained using the expansion of 1 divided by an irreducible polynomial [15]. Thus