ion and out-of-phase autocorrelation values from the set - 1, -1 + 24 and -1 - 2L12. Adding the long-period m-sequence to this set gives 2L12 sequences with a Pmax of 1 + 2L12. Evaluation of the Welch bound (5.241) at J = 2L12 and N = 2r 1, gives >(2 -2.242+1)122 = 1 for some r, )<r<1. Since the inequality is strict and since Pmax must be odd for binary sequences with odd periods, the Welch bound for this binary case reduces to (5.291b) The small set of Kasami sequences achieves this bound with equality, and, therefore, is optimal in the sense of minimizing the maximum absolute cor- relation parameter Pmax Theorem 5.20 can be used to verify that none of the compound sequences in the Kasami set is balanced. 5.7.5 A Transform-Domain Design Philosophy The linear designs of the previous section have excellent periodic correla- tion properties, but short linear spans. A transform domain design approach will now be described which results in a NLFFL design producing balanced binary sequences with optimal correlation properties and longer linear span [32],[42]. Consider two functions, r(x) and s(x), mapping the elements of GF(2d) into +1 and - -1. These functions can be viewed as representing NLFFL functions Direct-Sequence Multiple-Access Designs operating on a Galois m-sequence generator with characteristic polynomial ma(z), whose d-stage register contents p , r(n) represent an (see (5.127)). The resulting binary (+1 and -1) sequences, {r(a")} and {s(a")}, have peri- odic cross-correlation at shift T given by 2d - 1 (5.292) where y = . The inner product of r(yx) and s(x) over x E GF(2d) in (5.292) can be stud- ied in a transform domain using any inner product-preserving transform. Two closely related transforms will be considered for this purpose, the first being the trace transform which is defined as (5.293) for all 1 E GF(2). The following properties of the trace transform can be verified: (a) Inversion theorem: (5.294) (b) Multiplicative shifting theorem: For y # 0, s(x) = r(yx) for all X iff s(A) = r(y-1) for all