periodic autocorrelation function Paa(T) of {an} is given by = {24 T=0 mod 2M - 1 T # 0 mod 2M - 1. (5.190) Proof. It follows immediately from trace function linearity that (5.191) Let T be the smallest power of a yielding an element of GF(2), i.e., T=24-11 (5.192) and express the index n in (5.191) as n=no+nT, (5.193) where 0 VI no, T and 0 V n1 V 2 - 1. Using the linearity of the inner traces in (5.191) gives T-12'-2 (5.194) where (5.195) Since r is relatively prime to 2 - 1, it follows that art is a primitive element of GF (21), and QrTn, takes on the values of all non-zero elements of that field as n1 varies over its range. Hence, by including the zero element of GF(2) in the sum of (5.194), one arrives at (5.196) BeGF(2') When S(T,No) is not zero, the inner sum vanishes because half the exponents are zero and half one, by trace Property T-4. Let No(T) denote the number of values no in the range 0 < no < T for which S(T,no) is zero. Then (5.196) reduces to (5.197) Non-Linear Feed-Forward Logic Since r is relatively prime to 2 - 1 and therefore has an inverse modulo 2 - 1, it follows that Tr3"((a" - 1)a") = : 0. (5.198) When aT - 1 is zero, the right-hand equation in (5.198) is satisfied for all no. If - 1 is not zero, then Lemma 5.1 can be applied with (5.198) to deter- mine No(T). Thus, T = 0 mod 2 M - - 1 No(T) (5.199) and the result follows by substitution of (5.199) and (5.192) into (5.197). The linear span of a GMW sequence can be evaluated exactly by finding the number of terms in its powers-of-a representation and applying Theorem THEOREM 5.13. Let {bn} be a GMW sequence whose elements are given by (5.200) where a is a primitive element of GF(2M) and r, 0 <r<2 - 1, is relatively prime to 2 - 1. Then the linear span L of {bn} is given by L = J(M/J)", (5.201) where W is the number of ones in the base-2 representation of r. Proof. The exponent r can be written as = =2, (5.202) where the ji's are distinct integers in the range 0 < ji <J for all i. Hence, (5.203) Since the inner trace is a sum of elements f