trace properties (5.123) and (5.125) on (5.268) gives Paa(T) = (1)T((( (-1)Th (5.269) veGF(2L/2) the latter step using the fact that V and v2 have the same trace. The interior sum over V in (5.269) is zero by tace property T-4 unless the coefficient of V is zero, in which case the sum is 2L12. Furthermore, the coefficient of V in (5.265) is zero for exactly one value of u, namely (5.270) and it follows that Paa(T) -1+242(-1)T(BuT(3 (5.271) Theorems 5.17 and 5.18 will now be applied to the design of large sets of sequences with good periodic correlation properties. 5.7.4 Linear Designs Two roots-of-unity m-sequences, {an} and {a'n}, with corresponding sequences {bn} over GF(p) and {b'n} over GF(p'), possessing known cross-corelation properties, can be used to construct a larger set of sequences with easily determined correlation properties. In this general situation, the periodic cross-correlation of the roots-of-unity m-sequences is defined as (5.272) where p and p' are primitive complex p-th and p'-th roots of unity respec- tively, and (5.273) L and L' bieng the linear spans of the corresponding m-sequences. The fol- lowing theorem relates the cross-correlation properties defined in (5.272) to those of a larger collection of sequences. THEOREM 5.19. Let {bn} and {b'n} be the m-sequences described above and (5.274) Direct-Sequence Multiple-Access Designs Then the set of N sequences, (5.275) have periodic cross-correlation (over Nm symbols) taking on values -Nm/(P'L' - 1), -Nm/(p) - 1) and the cross-correlation values calcu- lated in (5.272). Proof. The cross-correlation of {a(}}} and {a(}}} can be simplified imme- diately, using the shift-and-add property of m-sequences. (5.276) Case (a): Assume T +j k # 0 mod pL and T # 0 mod p'L' - 1. Then from (5.63), = Paa(k+T'(T+j-k)- - T"(T)). (5.277) Case (b): Assume T + j - k # mod p L - 1, and T=0 mod p'L' - 1.In - this case, b'n cancels in the exponent of (5.276) and = = 1). (5.278) Case (c): Assume T + k = 0 mod pl - 1, and T # 0 mod p'l' - 1. As - in case (c), (5.276)