 reduces to (5.279) Case (d): Assume T + j k = mod pl - 1, and T = 0 mod p'L' - 1. Pik(T) = N m. = (5.280) This last case can occur only if T = C1(p'L' - 1), T+j-k=C2p-1), = (5.281) whence (5.282) Since the right side of (5.282) is a multiple of Nd, the allowed range of j and k excludes the occurrence of case (d), except when j = k and T = 0. When p = p', there is a common mathematical structure available for describing the sequences in Theorem 5.19. The characteristic polynomial of Pseudonoise Generators {bn} is a minimum polynomial ma(z) degree L over GF(p) with a a primi- tive element of GF(p). (5.283) Likewise, mB(z) of degree L' over GF(p) is the characteristic polynomial of {b'n}, where B is a primitive element of GF(p)'), and we assume that ma(z) # mB(2). (5.284) The generator for a sequence {a(i}} can be organized with p-ary shift registers as shown in Figure 5.15. This generator form is equivalent to a sin- gle L + L' stage shift-register generator producing a sequence over GF(p) with characteristic polynomial ma(z)mg(z). Let bk denote an L-tuple of {bn} beginning at the k-th symbol, and simi- larly denote an L'-tuple from {b'n}, beginning at the k-th symbol, by b'k. Hence, the states of the two Fibinacci generators in Figure 5.15 can be spec- ified by one such L-tuple and one such L'-tuple from the component sequences, provided the registers are not loaded with all-zeroes vectors 0. The results of various register initializations on the generator's GF(p) out- put sequence {cn} are shown in Table 5.17. The notation {b,i}}} has been used here to describe the GF(p) form of the output sequence, and, hence, when neither register is initialized with a zero vector, (5.285) for all j and n where p is a complex primitive p-th root of unity. The set of sequences {cn} for all i and k, enumerated in Table 5.17 forms a linear space. Specifically, if {cn} and {c'n} both satisfy the linear recursion having characteristic polynomial ma(z)mg(z), then SO do sequences of the form {dcn -ec'}} for any choice of d, e GF(p). 