seudonoise Generators together as a DS-SSMA signal set. We approach this question by consider- ing the cross-correlation between a roots-of-unity m-sequence {an} and its r-th decimation {am}, the latter being another m-sequence whenver r is rel- atively prime to the period of {an}. Let {Tr2(a")} denote a binary m-sequence, a being a primitive element of GF(2), and let {an} be the corresponding roots-of-unity m-sequence. The periodic cross-correlation between {an} and its r-th decimation {an}, a'n = arn for all n, is given by (5.246) Since a is primitive, an scans through the non-zero elements of GF(24) as n varies in (5.246), and, after compensating for the insertion of the zero ele- ment into the calculation, (5.246) leads to A,(y)1Pa(1T = (5.247) x EGF(24) where y = . Analytical simplifications of (5.247) have been achieved only for certain values of r. A comprehensive survey of these results, including tabulations of computer searches is given in [37]. One particularly good result will be derived from the following lemmas. LEMMA 5.2. Let r = 2 + + 1, e = gcd(2k, L), (5.248) and assume e divides k. Then A,(y) takes on three values: 0, 2(L+e)/2, and -2(L+e)/2 with 0 occurring for 2L - 2L-e different values of y. Proof. Squaring and simplifying (5.247) gives [A,(y)]E(-1)T)xz E (-1)Tr(t( weGF(24) (5.249) Equality (1) results from the substitution of Z = W + x and the form of r, and equality (2) employs trace linearity and the fact that wx2 and xw2 have the same trace since they are conjugates. As X varies over GF(2), the trace in the inner sum is 0 and 1 equally often (see trace Property T-4) provided that w2k + w2 is not zero. Hence, the Direct-Sequence Multiple-Access Designs sum on x is 0 or 24, and (5.249) reduces to [1,(y)]2=2(-1)Tr(w) (5.250) where N = = = {w:weGF(22k),weGF(24)}. (5.251) The intersection of GF(22) and GF(24) is GF(2), where e is the gcd (2k, L). The order of every W in GF(2) divides 2e - 1, and, therefore, assuming e divides k, w = www. = (5.252) where C is an integer. Since the traces of W 