De Bruijn Sequences from Symmetric Shift Registers

We consider the symmetric Feedback Shift Registers (FSRs), especially a special class of symmetric FSRs (we call them scattered symmetric FSRs), and construct a large class of De Bruijn sequences from them. It is shown that, at least O(2 n−6 2 ) De Bruijn sequences of order n can be constructed from just one n-stage scattered symmetric FSR. To generate the next bit in the De Bruijn sequence from the current state, it requires no more than 2n comparisons and n+ 1 FSR shifts. By further analyse the cycle structure of the scattered symmetric FSRs, other methods for constructing De Bruijn sequences are suggested.