uce the output sequence over GF(p). Note that the structure of this NLFFL is dependent on the sequence to be generated, i.e., the index j, and is generally rather com- plex. Insight into the balance properties of a sequence {b,}}} in a linear design can be obtained by relating balance to correlation. Let Nb be the number of occurrences of the symbol b in one period of {bi}} and note that (5.286) beGF(p) Knowledge of the periodic cross-correlation values on the left side of (5.286) imposes constraints on the possible values of N and vice versa. The clear- est example of this relation occurs when p=2, = and is summarized in the following. THEOREM 5.20. Let - 1, be the compound sequences of a binary linear design, based on component m-sequences {bn} and {b'n} with corresponding roots-of-unity m-sequences {an} and {a' }} respec- tively. Then the number of compound sequences in the binary linear design, having N ones in one period, is given by the number of shifts j, Direct-Sequence Multiple-Access Designs 0 j, N, for which where Nm and N 1 are defined in (5.273) and (5.275) with p = p' = 2, and L and L' are the linear spans of the component m-sequences. Note that Nm - 2N1 is the number of zeroes in the binary sequence {b,}}} minus the number of ones, and, hence, Paa'(j) can be interpreted as the imbal- ance in {bi}}} Example 5.14 (Gold sequences). In a 1967 paper [35], Gold suggested com- bining two binary m-sequences with identical periods to produce a set of SSMA codes with good periodic correlation properties. Gold considered a linear design in which two component sequences, each with linear span L, were employed, one with characteristic polynomial ma(z), and the other with characteristic polynomial ma(z), where (5.287) with x denoting the integer part of X. This choice of r satisfies the condi- tions of the cross-correlation Theorem 5.17 with e = 1 for L odd and e = 2 for L even. The combining process of Theorem 5.19, in this case with p = p' = 2 and Nm = N =2L- - 1, yields a binary linear design with all cross-c