On the cycle structure of certain classes of nonlinear shift registers

When m = qt, g(x_t+1, x_2t+1,…, x_(q?1)t+1) is a linear combination of only odd (or only even) elementary symmetric functions, then every cycle of the nonlinear shift register with feedback function f(x₁, x₂,…, x_m) = x₁ + g(x_t+1, x_2t+1,…, x_(q?1)t+1) has a minimal period dividing m(q+1). It is also shown that when g is derived from a cyclic code with minimum distance ?3, every cycle of this shift register has a minimal period dividing m(q + 1).