Period and Complementarity Properties of FCSR Memory Sequences

In this correspondence, we investigate feedback with carry shift register (FCSR) memory sequences. For an -sequence generated by an FCSR with connection integer and initial memory , we prove that for , where and is the memory sequence. Generally speaking, the period of a memory sequence is a factor of that of the FCSR output binary sequence. We show there are a large number of connection integers, with which an FCSR can generate sequences that have the same period as their memory sequences, especially including all connection integers for -sequences.