n of a complementary basis. A compact description of the preceding development is given in the following result. THEOREM 5.22. Let a be a primitive element of GF(2), d divisible by 4, and let x represent the contents of a Galois-configured m-sequence generator having the minimum polynomial of a over GF(2) as its characteristic poly- nomial, i.e., x = X x. (5.343a) Let E0 be any element of GF(2d) that is not contained in a smaller field, and let {1, , be a basis for GF(2d)2 over GF(2). Let M be a d/2 X d matrix whose i, j-th entry Mij is given by (5.343b) and let st be any d-dimensional vector not contained in the linear subspace spanned by the rows of M. Then the 2d12 NLFFL functions of the form (5.343c) indexed by Z E Vd/2, where f2.) is a bent function of the form (5.320), pro- duce balanced sequences with periodic cross-correlation and out-of-phase autocorrelation bounded in magnitude by 1 + 2d12. Example 5.16. Consider a primitive polynomial 212 26 with root a in GF(4096), which serves as the characteristic polynomial of a Galois LFSR. This register's state sequence has period 4095, and its contents x represent the field element x in the basis {a: i=0,1,...,11} as indicated in (5.343). In this notation, the set {Q65: i=0,1,...,5} is composed of ele- ments having order dividing 63, Q65 being primitive in GF(64), and, hence, these elements form a basis for the subfield GF(64). One possible choice for the element E0, which must be outside GF(64), is a. With the choices just stated, the 6 X 12 basis reduction matrix M has ij-th entry (5.344) Direct-Sequence Multiple-Access Designs resulting in the matrix 000000000010 001100010111 100001110000 (5.345) 101001100100 110010100001 011001001001 A suitable choice for the vector st, which must be outside the row space of M, is = (000000000001). (5.346) Employing a bent function of the form described in Theorem 5.21, operat- ing on Mx, with the arbitrary function g() being a three-input AND gate, one bent-sequence generator design is shown in Figure 5.16. The set of bent 