inning with the k-th integer. Then for T # 0 mod q1 - 1, the Hamming distance between i and i1++ is given by (5.69) Pseudonoise Generators Proof. The proof of this result follows from Property R-3 which establishes in=in+t (5.70) with a count of the latter event evaluated in Property R-2. Using a variation of Plotkin's bound, developed by Lempel and Greenberger [11], it can be proven that the integer sequence {in} of (5.67) possesses optimal Hamming distance properties. THEOREM 5.4. Let {an} be a sequence of period N, composed of symbols from an alphabet A, and let aj denote the N-tuple of consecutive symbols from {an}, beginning at the j-th symbol. Let (5.71) and determine the integers a and B from NaA+B,OSB<A. (5.72) (5.73) Proof. Certainly the minimum value Hmin of Hamming distance between {an} and its cyclic shifts must be less than the average value Havg of the same quan- tities. Now (5.74) Let N denote the number of times that the symbol "a" occurs in one period of {an}. The number of times that h(am,An+7) is zero as T varies over its range is simply Na.-1.Hence, (5.75) Removing the dependence of (5.75) on the set of integers {Na} yields the bound (5.76) Statistical Properties of M-Sequences where {Na} is a set of non-negative integers summing to N. It can be verified that the right side of (5.76) is maximized by making the integers N as nearly equal as possible. That is, the maximizing choice for {Na} contains the inte- ger a + 1 a total of B times and the integer a a total of A - B times. Substitution of these values into (5.76) and simplification using (5.72) yields the final result. Evaluation of the bound (5.73) for the parameters N = q - 1, Al=q, (5.77) whence B=q'-1, (5.78) gives the bound (5.79) Property R-4 indicates that {in} satisfies this bound with equality and, there- fore, possesses the desirable attribute of being uniformly maximally distant from its proper cyclic shifts. 5.4.4 Correlation Properties of Derived Complex Roots-of-Unity Sequences The case in which the field size q is a prime p,