On a Homomorphism of the de Bruijn Graph and its Applications to the Design of Feedback Shift Registers

A homomorphism of the de Bruijn graph that maps a graph of order n onto one of order n-1 and its applications to the design of nonsingular feedback shift registers are discussed. The properties preserved under this mapping suggest a new design technique whose main advantage is due to the fact that the problem of designing a desired n-stage shift register may be reduced to a problem of order n-1 or less. Among the results obtained is a recursive formula for a feedback function that generates a cycle of maximum length.