rom a field of characteristic 2, all cross-product terms disappear when the trace is squared and (5.203) reduces (5.204) Pseudonoise Generators where K is defined by (5.184) and (5.205) The time coefficients c(k,r) for the representation of the nonlinear sequence over GF(2), as expressed in (5.205), are all less than 2M - 1 (the order of a). Since the exponents of 2 in (5.205) are distinct modulo J, it is easily ver- ified that no two distinct k vectors produce the same time coefficient, c(k, r) and, therefore, no terms in the w-fold sum in (5.204) can be combined. When expanding the outer trace function in (5.204), cross-product terms again must disappear, and (5.206) m=0 k =0 Therefore, the time coefficients in the powers-of-a representation of {bn} are simply c(k,r)2m. To determine whether or not these coefficients are distinct modulo the order of a, solutions k k2, m1, m2 to the equation k1,r)2mc(k2,r)2mmod 2M - 1, (5.207) must be sought. Since 2 - 1 divides 2M - 1, any solution to (5.207) must also satisfy the same equation modulo 2 - 1. But c(k,r) modulo 2 - 1 is simply r, and (5.207) modulo 2 - 1 can be rewritten as 2m2) = 0 mod 2 - 1. (5.208) However, r is relatively prime to 2 - 1 and, hence, in the allowed range of m and m2, (5.208) exhibits only the solution m1 = m2. Therefore, the only solution to (5.207) is the identity k = k2 and m1 = m2, and all time coeffi- cients in (5.206) are distinct. Application of Theorem 5.9 and counting the terms in (5.206) yields the final result. Example 5.12. A GMW sequence of period 63 is defined by (5.209) where 26 + is the minimum polynomial of a over GF(2). One simple plan for mechanizing a generator for {bn} is to construct a gen- erator for the m-sequence Tro (a") and use a ROM to complete the map- ping to GF(2). The elements of GF(8) are 0 and a9i, i = 0, 1, minimum polynomial of a over GF(8) is easily determined (see Appendix 5A.8) to be z2 + 54z + a9. A block diagram of the generator employing GF(8) arithmetic is shown in Figure 5.13. The actual mechanization t