On the cycle structure of certain classes of nonlinear shift registers (Q795051)

Let \(g(x_{t+1},x_{2t+1},...,x_{(q-1)t+1})\) be a linear combination of only odd (or only even) elementary symmetric functions and \(m=qt\). The two main results of the paper are the following: 1. Each cycle of a nonlinear shift register with a feedback function \(f(x_ 1,...,x_ m)=x_ 1+g(x_{t+1},x_{2t+1},...,x_{(q-1)t+1})\) has a minimal period dividing \(m(q+1)\) (theorem 7). 2. If g is obtained from a cycle code with minimum distance \(\geq 3\), the period of any cycle of this register must also be a factor of \(m(q+1)\) (theorem 10).