 functions with six terms and one function with two terms, the linear span L of {bn} is deter- mined to be 38. The expansion (5.164) indicates that the characteristic polynomial Q(z) of the NLFFL output sequence {bn} is Q(z) = (5.165) The factors of Q(z) can be determined from knowledge that ma(z) is and the techniques of Appendix 5A.8, to give 83225 +z2+z+1. (5.166) As a check, it can be verified that the output sequence shown in Figure 5.10(b) satisfies a recursion based on the characteristic polynomial Q(z). 5.6.2 Key's Bound on Linear Span Suppose that a binary NLFFL function is put in Reed-Muller canonic form, i.e., described as a sum of products of its inputs. Let's investigate the poten- tial number of terms contributed to the powers-of-aexpansion of the logic's output by one product term containing J factors, each factor being the con- tents of some memory element in an m-sequence generator. Assuming the characteristic polynomial of the m-sequence generator is ma(z), the trace function Tr(y/a") can be used to represent the j-th factor (sequence) in the product Pn, giving (5.167) Pseudonoise Generators where and a are elements of GF(2M), M being the number of memory elements in the m-sequence generator and also the degree of ma(z). (This representation is valid for a linear m-sequence generator in any configuration.) Converting the product of sums in (5.167) to a sum of products gives (5.168) where (5.169) (5.170) the latter sum being modulo the multiplicative order of the element a. The base 2 representation of c(i), thus, is limited to a binary M-tuple, and (5.170) indicates that at most J of these M symbols can be ones. Since each of the integers ij in (5.170) ranges independently between zero and M - 1, it fol- lows that the number MNJ of distinct values that c(i) can assume is given by the number of binary M-tuples with at most J ones, excluding the all-zeros n-tuple. (5.171) Two different products (AND gates), with distinct sets of J inputs from the memory cells of an m-sequence generator, potentiall