 2L-e-1 2(L-e-2)/2 values of T (5.263) where OST<24- - 1. Final determination of the counts shown in this theorem can be made after showing that Paa(T) = 1. Information concerning other pairs of m-sequences with good periodic cross-correlation is given in Table 5.16. It is assumed in this listing that r is relatively prime to 2L - 1, and e is the greatest common divisor of L and 2k. Note that while Welch's proof of the second listed result is unpublished, it is the m = 3 case of the third listed result. Each of the entries in Table 5.16 gives rise to peak cross-correlation magnitudes on the order of 2(L+1)/2 when L is odd, and 2(L+2)/2 when L is even. However, this table does not list all such cases. Direct-Sequence Multiple-Access Designs Table 5.16 Maximum absolute value of the cross-correlation between m-sequences with characteristic polynomials ma(z) and ma(z) a a primitive element of GF(2). maxPad[pr](T) Comments 2(L+e)/2 + 1 2(L+e)/2 (Welch, unpublished) (2mk + 1)/2k + 1) 2(L+e)/2 m odd, [37] 2(L+2)/2 1 2(L+2)/2 1 L = 0 mod 4, [37] 2L-1 1 < 2(L+2)/2 Periodic cross-correlation calculations can be carried out analytically on m-sequences of differing periods, sometimes with good results. This is illus- trated in the following theorem and proof. THEOREM 5.18. Let a be a primitive element of GF(24), L even, and let r = 2 4 2 - 1, (5.264) being a primitive element of GF(24). Then the roots-of-unity m- sequences of periods 2L - 1 and 2L12 1, with elements defined by = (-1)Tr2(a") = (5.265) (5.266) respectively, have periodic cross-correlation (over 2L - 1 sequence ele- ments) values -1 2 2 at all shifts. Proof. The cross-correlation between {an} and {an} at shift T is given by (5.267) x EGF(24) where X = an, y = , and the field element 0 has been added to the sum range. The elements of GF(24) can be represented in the form x = uß+v (5.268) where and V are elements of GF(2L/2) and B is the root of an irreducible quadratic polynomial over GF(2L/2) (see Appendix 5A.5, Theorem 5A.9) Pseudonoise Generators Then, using 