0) has linear span (MIJ), where W is the number of ones in the binary representation of a mod 2 - 1. The linear span of the sequence represented by the right side of (5.230) is MIJ, since a is also a primitive element of GF(2M). Hence, equality requires that W be 1 and, therefore, (5.223) is necessary. Substitution of (5.223) into (5.227) results in (5.222) as a necessary requirement. It is easily verified that these necessary conditions are also sufficient. Hence, the number NGMW of cyclically distinct GMW sequences, which can be constructed by this method for a fixed M and J, is given by Namw = N,(M)N,(1), (5.231) where Np(d) denotes the number of primitive polynomials of degree d over GF(2), or equivalently the number of cyclically distinct m-sequences of lin- ear span d over GF(2) (see Section 5.3.4). Example 5.13. Consider a design in which the ROM size constrains J to be 7. Since 27 - 1 is the prime number 127, all binary 7-tuples, except 0000000 and 1111111, are radix-2 representations of numbers relatively prime to 127. One set of acceptable choices of r, with cyclically inequivalent radix-2 Pseudonoise Generators Table 5.15 Design parameter tradeoffs for GMW sequences with a constraint J = 7. M = 28 (w)/J r values NGMW(w) NGMW(w) 4741632 3, 5, 9 14224896 7, 11, 13, 19, 21 23708160 15, 23, 27, 29, 43 23708160 31,47,55, 14224896 4741632 28672 representations, is shown in Table 5.15, which contains a total of Np(7) entries with (w)/J entries for each value of W. This table also indicates the number NGMW(w) of GMW sequences constructed with weight-w values of r, along with their linear span L, for two possible values of memory size M. 5.7 DIRECT-SEQUENCE MULTIPLE-ACCESS DESIGNS 5.7.1 A Design Criterion In many situations, more than one SS code generator is required to operate simultaneously within the same locale and bandwidth, i.e., in a multiple- access (MA) environment. The distinguishability of different SS signals by receivers participating in the SSMA scheme depends on many factors, including the SS se