Relations Between Pn Cycles and Stable Feedback Shift Registers

In this paper a relationship among distinct Pⁿ⁺¹ cycles, distinct stable maximum transient feedback shift registers of order n+1, and stable feedback shift registers of order n will be presented. In the course of the discussion, two algorithms will be introduced. The first will provide a one-to-one mapping between distinct stable feedback shift registers of order n and distinct stable maximum-transient feedback shift registers of order n+1. The second will provide a one-to-one mapping between distinct maximum transient feedback shift registers of order n+1 and distinct Pⁿ⁺¹ cycles. As a corollary to these relationships, an enumeration of the stable feedback shift registers is obtained.