 usually 2 in practice, deserves added attention. In this case, GF(p) arithmetic is modulo p arithmetic, and the field GF(p) is composed solely of integers 0, 1, p - 1. By viewing these integers in GF(p) as being identical to their counterparts in the real numbers, the following mapping from bn in GF(p) to a complex p-th root of unity can be constructed: an=pbn, (5.80) where p is a primitive p-th root of unity in the field of complex numbers, e.g., p = exp(i2/p). (5.81) (In the special case when p = 2, then p = -1 and bn E GF(2) = {0, 1} is mapped into an E {1, -1}.) Consequently, additional modulo p of two ele- ments bm and bn from GF(p) in the domain of the mapping (5.80) is iso- morphic to complex multiplication of two elements in the range of the mapping, i.e., aman = pbmtbn, (5.82) with addition of integers in the exponent being both real and modulo p since p is the p-th root of unity. An m-sequence {bn} of elements GF(p), satisfying the linear recursion (5.83) Pseudonoise Generators can be mapped by (5.80) into a sequence {an} of complex p-th roots of unity satisfying the multiplicative recursion (5.84) Note that {an} generally does not satisfy a linear recursion of degree L over the complex numbers. However, the resulting complex sequence {an} often retains the name m-sequence. Since the m-sequence {an} consists of complex numbers, it is possible to evaluate its periodic autocorrelation function Paa(T), defined by (5.85) where ( ** denotes conjugation. The nearly ideal periodic correlation func- tion of m-sequences is described in the following result. Property R-5. Let {an} be a complex m-sequence of period N, composed of p-th roots of unity. The periodic correlation Paa(T) of {an} has the form T=0mod N (5.86) Proof. The case T = 0 mod N is obvious. Note that N = pL - 1, L being the linear span of the corresponding m-sequence {bn} over GF(p), and by R-2 the number of occurrences of any specified non-zero symbol in one period of {bn} is pL-1. Hence, for T # 0 mod N, [by (5.80)] [by R-3] [by R-2] = -1. (5.8