 where each polynomial Qj(z) is irreducible over GF(q), and the linear span L of {bn} is given by (5.46) The term irreducible over GF(q), when applied to a polynomial A(z) over GF(q), means that A(z) cannot be factored into a product of polynomials over GF(q). Hence, the factorization described in (5.45) is similar to the decomposition of an integer into prime factors. The factorization (5.45) indicates that the rational representation (5.16) of the formal power series B(z) can be expanded by partial fractions. (5.47) Therefore, the sequence represented by B(z) also can be generated by sum- ming the component sequences represented by (5.48) the j-th sequence having characteristic polynomial Qf;(z) and initial condi- tion polynomial P(z). Let's define the period of the j-th sequence to be Nj. Then the period of the composite sequence {bn} is given by N = 1cm(N1, N2, N 1), (5.49) where 1cm() represents the least common multiple of the listed integers. Example 5.6. The sequence of Examle 5.4 was represented by where P1(z) and P2(z) are polynomials of degrees at most 1 and 3, respec- tively. Solving for these polynomials gives the component generators Memory-Efficient Linear Generators Figure 5.8. An LFSR generator for the sequence of Examples 5.4 and 5.6. (5.51a) (5.51b) The first generator produces the period 2 sequence while the second generator outputs the period 15 sequence 101011001000111. Extending these sequences periodically and summing gives the period 30 sequence of Example 5.4. The Galois-configured, component-structured, LFSR shown in Figure 5.8 can be initialized to have an output identical to the generators of Figure 5.7. 5.3.2 Maximization of Period for a Fixed Memory Size The fundamental memory constraint (5.5) indicates that the maximum period achievable by an q-ary generator with M memory elements is qM. This maximum period is reduced by 1 if the generator uses linear feedback, because the zero state is self-perpetuating, i.e., an LFSR initiated with zeros produces the uninteresting all-zeros seque