Algebraic theory of flip-flop sequence generators

The problem of constructing linear shift registers with a minimum number of adders has provoked interesting research on the theory of trinomials over the field with two elements. Each adder which can be eliminated significantly increases the speed at which the sequence can be generated, and linear shift registers corresponding to trinomials have only one adder. In this paper we describe a class of sequence generators employing J\2-K flip-flops in place of the usual delay elements, and which require no adders or additional gating. J\2-K flip-flops operate at a speed comparable to that of delay elements. If n is the number of flip-flops, then for n = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 15, 17, and 18 a sequence of period 2{sun} \3- 1 can be generated. This sequence is linear and has the well-known randomness and correlation properties. A table in the final section of this paper gives the periodic structure for all n ≦ 19.