 1. (5.295) (c) Parseval's relation: (5.296) x EGF(2d) 1 E GF(2d) (d) Additive shifting theorem: s(x) = r(x+y) for all X iff s(A) = r(A)(-1)Tr(x) for all 1, (5.297) all 1 iff s(x) = r(x)(-1)Tr(xy) for all X. (5.298) Another equivalent inner-product-preserving transform is defined after a one-to-one linear mapping of GF(2d) onto the space Vd of d-tuples with elements in GF(2). Specifically, let B1...., B be an arbitrary basis for GF(2), and let Y1,..., Yd be another basis for GF(2) with the property that (5.299) Pseudonoise Generators Two bases satisfying (5.299) are called complementary. It can be verified [42] that every basis for GF(2) has a complementary basis. Let x and l be arbi- trary elements of GF(2), represented in complementary bases by (5.300) Coordinate values can be determined by applying (5.299) and trace linear- ity to (5.301) to yield (5.301) for all i. These same relations allow Tr(Ax) to be interpreted as an inner prod- uct of vectors of coefficients from the expansions (5.300). (5.302) The mapping from x E GF(2d) to XEVd, can be used to define R(x) 4 (r(x) (5.303) for all X in GF(2d), substitution of (5.302) and (5.303) into the trace trans- form definition (5.293) gives (5.304) The right side of (5.304) defines the Fourier transform R(A) of the function R(x). The inversion theorem, Parseval's relation, and additive shifting theo- rem for the Fourier transform are easily derived from the corresponding results for trace transforms, in view of (5.304). Both transforms will be used in the design approach to follow. Returning to the periodic cross-correlation computation of (5.292), we now develop a bound on cross-correlation. Applying Parseval's relation (5.296) gives (5.305) Therefore, Rs(t)|1y13(A), (5.306) leys,NIs where the sum over 1 in (5.306) has been limited to those 1 for which the corresponding term is non-zero, i.e., (5.307) (5.308) Parseval's relation also implies that (5.309) 1 E GF(2d) x EGF(2d) Direct-Sequence Multiple-Access Designs which yields a bound on the trace transform, name