ynomial Q(z). (5.64) and the difference of {bn++} and {bn} therefore satisfies the same recursion. - - bn-i) for all n. (5.65) Statistical Properties of M-Sequences Since {Sn} represents the state sequence for a Fibonacci generator produc- ing {bn}, then the initial state S'n for {bn++-bn} - is (5.66) This initial state S' is non-zero if, and only if, T is not a multiple of the period q L - 1 of the state sequence {Sn}. When S' is non-zero, it is a state S1++'(7) of the Fibonacci generator of {bn}, and, hence, {bn++'(7)} is the result of the recur- sion (5.64) operating on this initial state. When the symbol field has characteristic 2 as it does in the case of binary (0 or 1) sequences, subtraction and addition are identical operations, and Property R-3 is usually called the shift-and-add (or cycle-and-add) property. It also is true that the sum of an m-sequence and its proper cyclic shift is another shift of the m-sequence, but the form (5.63) is more useful in the general results to follow. 5.4.3 Hamming Distance Properties of Derived Real-Integer Sequences Sometimes an m-sequence generator over GF(q) is used to produce a pseudo-random sequence of integers between 0 and q' - 1 by viewing the n-th J-tuple in the m-sequence {bn} as the base q expan- sion of the n-th integer in in the output integer sequence. (5.67) Some applications using integer sequences like {in} require that the sequence be as distinct from its cyclic shifts as possible. One measure of this quality is the Hamming metric which counts the number of positions in which two N-tuples x and y differ. This metric takes the form (5.68) where h(x,y) is zero if x and y are identical, and one otherwise. The follow- ing attribute indicates that {in} behaves much like a random sequence, with symbol matches between it and its cyclic shift occurring at a rate very close to one in every q' tests. Property R-4. Let {bn} be an m-sequence over GF(q) with linear span L, and let {in} be formed according to (5.67). Let ik denote one period of the sequence {in}, beg