inear subspace of GF(2d) with basis My. Substituting (5.328) into (5.326) and interchanging the order of summa- tion gives (5.330) x EGF(2d) The sum over x is non-zero if and only if the coefficient of X (in the trace function) is zero. Hence, Fz(1) is non-zero if and only if o + l is in the range of e(.). Denoting the inverse of e on the range of l by e-1, it follows that for o otherwise. + 1 erange(l) (5.331) Since F 2 is the transform of a bent function, F(A) takes on two values, 0 and 2, as 1 varies over GF(2d), thereby satisfying design properties P-1 and P-4 of the previous section. One purpose of the added term Tr(ox) in the design of fz(x) is now apparent: The element o can be chosen to make rz(0) = 0, thereby guaranteeing that the bent sequences are balanced. Two applications of Parseval's relation can be used to verify the orthog- onality of functions in B. 1 E GF(2) = 0 for Z # W. (5.332) Thus, design objective (Property P-3) is satisfied, guaranteeing good "in- phase" cross-correlation properties of bent sequences. All requirements of the design procedure (Properties P-1 to P-4) have been satisfied except Property P-2, i.e., the minimization of YSNS. In this design, the set I is defined from (5.331) as S{1:o+derange(l)} (5.333) Since o is a constant and the range of l contains 2k points, S contains 2k points, and (5.314) indicates that a good design may at best achieve ySnS<< for all y E GF(2). (5.334) Direct-Sequence Multiple-Access Designs The final design step is performed by choosing an appropriate basis for the range of e(.) SO that (5.334) is satisfied. let range(l){8e0:deGF(2)} (5.335) where E0 is any element of GF(2d) that is not contained in a smaller field. Then S in (5.333) may be represented as (5.336) We will now demonstrate that (5.334) is satisfied. Viewing GF(2d) as an extension field of GF(2) which can be represented as a two-dimensional vector space over GF(2) (see Appendix 5A.4), an arbitrary element y of GF(2d) can ber represented as y = y180 + yo, (5.337) where the basis for the v