Nonsingularity of feedback shift registers

In this paper, the multi-valued feedback shift register (FSR) is studied and a new approach is present to analyze its nonsingularity, number of cycles, and cycle synthesis. Firstly, the FSR is expressed in an algebraical form, based on which several necessary and sufficient conditions are given for the nonsingularity. Secondly, the structural matrix of FSR is defined, and a new method is introduced to determine the number of cycles with different lengths for arbitrarily given FSR. Thirdly, the problem on cycle decomposition and synthesis of an FSR is investigated, and some new results are obtained. Finally, an illustrative example is studied to support our new results.