nce. Thus, the most efficient LFSR generators must cycle through all possible non-zero states before repeating, and the period N of an LFSR's state sequence is bounded by N < q M - 1. (5.52) In Example 5.6, the four-stage component LFSR generator achieved its max- imum possible period of 15, while the two-stage generator did not; and the composite generator of six stages produced a period 30 sequence, nowhere near the upper bound of 63 dictated by (5.52). Pseudonoise Generators Assuming a Q(z) of the form (5.45), applying the bound (5.52) to the com- ponent sequences, and using (5.49) gives N = IIN, (5.53) It is clear from (5.53) that the fundamental bound (5.52) on N can be achieved if and only if all the above inequalities hold with equality. Equivalently, considering each inequality in turn, equality holds in (5.53) if and only if (a) the periods of the component sequences are relatively prime, (b) each component sequence achieves the bound (5.52) on its period for its memory size, and (c) the number J of component sequences must be 1. When the degrees of the component generator polynomials are large, then item (c) is not critical to the efficient use of memory in creating large periods. 5.3.3 Repeated Factors in the Characteristic Polynomial Consider now the special case in which the characteristic polynomial of a sequence over GF(q) is the q-th power of a polynomial over GF(q). Theorem 5A.10 of Appendix 5A indicates that Q (z) = (5.54) Hence, the coefficient list of Q (z) is simply where 0 0 represents a string of q - 1 zeros. A shift register governed by this form of generator polynomial will produce q interleaved sequences. That is, a symbol b in the composite output sequence {bn} is linked recursively only to the prior symbols bn-mq, m=1,2,3, = This line of reasoning can be iterated to yield the following result. THEOREM 5.2. Let N be the period of a sequence having characteristic poly- nomial Q(z) over GF(q). Then the period N' of the sequence having char- acteristic polynomial QP(z) is given by N' = q'N