Distributions Modulo Subgroups of GF(q)

We consider recursive sequences over GF(q) (the Galois field with q elements) generated by an irreducible polynomialfof degree 12. All roots of f have the same order e (called the exponent off) in the multiplicative group GF(qn)* = GF(q”) (0). Every nonzero recursive sequence generated byf is periodic with period e, and there are (up to translation) (4% 1)/e nonzerof-sequences [l], [2]. If e = q” 1 the polynomia1.f is primitive, and in a single period of the nonzero f-sequence zeros appear qn-l 1 times and every nonzero element appears q”-l times. This is easily seen because every n-tuple except all zero must appear exactly once in a period. If e < qn 1 the distribution of elements within each individual nonzero f-sequence need not be so nearly uniform. We shall show that for certain values of e the distribution of each f-sequence is uniform over the cosets of a certain subgroup of the multiplicative group of GF(q)*. Specifically this occurs when [e, q 11, the least common multiple of e and q 1, is q” 1.