roach, with = be successfully demonstrated in the next section. 5.7.6 Bent Sequences Rothaus [43] defined a function f(x) mapping the space V of binary k- tuples over GF(2) into V1 (GF(2)) to be bent if the Fourier transform of (-1) (X), namely (5.318) is 1 for all 1 in Vk. Although there are many families of bent functions [43]-[45], only one bent function form will be described here. THEOREM 5.21 (ROTHAUS [43]). Let (5.319) Let the function fz(x), mapping V2j onto V1, be defined as f2(x)xx2+g(x2)+zx (5.320) where g(x2) is an arbitrary function mapping Vi into V1.Then fz(x) is bent for every Z in V2j. Direct-Sequence Multiple-Access Designs Proof. The Fourier transform of f2(x) is given by F2(A)2(-1)2(x) X2EV; X (-1) X1EV; where, in the latter step, Z. and l have been appropriately partitioned into vectors of dimension j. The sum over x1 in (5.321) is zero if and only if x2 + Z1 + 11 is not zero, its alternative value being 2. Hence, )=(-1)8(211)(1)( (5.322) With regard to dimensionality, this design will now fix d=2k=4j, = (5.323) with g() an arbitrary function from Vj to V1, and f2.) a bent function mapping V into V1. Let L(x) define a linear mapping from GF(2) to Vk. The set B of NLFFL functions used to produce bent sequences is now defined to contain all functions of the form rz(x) = (5.324) where o is fixed and Z. E Vk. The trace transform of rz(x) is given by (5.325) x EGF(2d) Representing (-1)f in terms of its inverse transform on Vk gives (5.326) x EGF(2d) The linear mapping L(x) can be represented as - L(x) = Mx = M Tr(y1x) (5.327) Tr(Yax)] where M is a k X d matrix over GF(2), of rank k and x is the vector repre- sentation of the GF(2d) element x (see (5.300)-(5.301)). The inner product X"L(x) in (5.326) can be rewritten using (5.327) and the linearity of the trace function, SO that (5.328) Pseudonoise Generators where the adjoint l(l') of L(x) is e(A') = A"My, (5.329) and Y is the vector of basis elements Y1,..., Yd. Since the rank of M is k, l(X') is a one-to-one mapping from V onto a k-dimensional l