ents in GF(2) which map into zero by (5.216) is 2-1 (trace property T-4), it is easily verified that AA (21-1)k, for all k-tuples a. (5.218) Therefore, when a # 0 and hence 0 E A then all the terms in (5.217) are identical by (5.215), and = for a # 0. (5.219) A similar computation for a = 0 completes the proof. The next result indicates that all proper decimations and allowable choices of the exponent r yield distinct GMW sequences. THEOREM 5.15. Let {bn} and {cn} be GMW sequences whose elements are given by b, = Tr3'([Tr3"(a")]'') (5.220) (5.221) where r and S are integers relatively prime to 2J-1 - and less than 2 - 1, a is relatively prime to 2M - 1, and a is a primitive element of GF(2M). Then {bn} and {Cn} are cyclically equivalent if and only if r = 2 s mod 2' - - 1 for some O<j<J (5.222) a = 2m for some 0 m m M. (5.223) Proof. Define T, no, and n1 as in (5.192) and (5.193). Let B=a and y=aT = (a primitive element of GF(2). Then the linearity property of the trace gives {bn}=={cn++} (5.224) where (5.225) (5.226) Non-Linear Feed-Forward Logic The trace functions in (5.224) represent two m-sequences over GF(2), each with element index N1. These sequences are identical for some B if and only if y and yas have the same minimum polynomial over GF(2); hence, equal- ity requires 2jas mod 2 - 1, for some OSj<J (5.227) (see Theorem 5A.14 in Appendix 5A). Assuming (5.227) is satisfied, the fact that conjugate elements have the same trace (see 5.122) can be used to show that (5.228) and, hence, equality holds in (5.224) when k2 is identical to k2-j i.e., when Tr(a)r(aans (5.229) Substitution of (5.227) into (5.229) and raising each side of the result to a power equal to the multiplicative inverse of S modulo 2 - 1 yields the result (5.230) If the two sequences over GF(2), whose elements with index no are given by the right and left sides of (5.230), are cyclically equivalent, then their lin- ear spans must be equal. It can be shown via the techniques of Theorem 5.13, that the sequence defined by the left side of (5.23