On k-Stable Feedback Shift Registers

A state graph (SG) is a directed graph with exactly one arc issuing from every vertex of the graph. The degree of an SG is the smallest integer d such that at most d, arcs are entering any vertex of the graph. An SG is said to be k-stable if it contains k≤1 cycles of unit length (loops) and these are the only cycles of the graph. A k-stable SG with mn vertices and degree d is called a (k, m, n)-SG if m≥k, d and if the distance from any vertex to a loop of the graph is at most n.