Periodicity, correlation, and distribution properties of d-FCSR sequences

A d-feedback-with-carry shift register (d-FCSR) is a finite state machine, similar to a linear feedback shift register, in which a small amount of memory and a delay (by d-clock cycles) is used in the feedback algorithm (see [4, 5]). The output sequences of these simple devices may be described using arithmetic in a ramified extension field of the rational numbers. In this paper we show how many of these sequences may also be described using simple integer arithmetic, and consequently how to find such sequences with large periods. We also analyze the “arithmetic cross-correlation” between pairs of these sequences and show that it often vanishes identically. Finally we study the distribution properties of short sub-sequences of a d-FCSR sequence.