ely prime, then the smallest multiple of J that is a multiple of N is NJ, and the period of {cn} is N. In general, the decimation by J of a sequence with period N produces a sequence with period N/gcd(J, N). It should be clear to the reader that when gcd(J, N) is 1, the resulting rearrangement of sequence elements does not change the first order statis- tics of the sequence. A more interesting result does not change the first order statistics of the sequence. A more interesting result indicates that the set of correlation values of real number sequences is preserved under decimation by a number relatively prime to the sequence period. THEOREM 5.7. Let {an} be an arbitrary sequence of complex numbers with period N, and let {cn} be the decimation by J of {an}, where gcd(J, N) = 1. Pcc(T) = Pad(It). (5.138) Proof. Evaluation of the periodic correlation of {cn} gives (5.139) Pseudonoise Generators where (={k:Jn=k mod N, = (5.140) If two distinct choices of n, say n1 and N2, in the range 1 to N, yield the same value k in (5.140), then 1=JmmodN (5.141) or, assuming n1 N2, (5.142) where m is integer. However, this is impossible since N has no factors in com- mon with J, and n1 - N2 is less than N. Therefore, the elements of K are the N distinct integers from 1 to N. The trace representation gives considerable insight into the effects of dec- imation on m-sequences. Suppose that by Corollary 5.2 we have constructed the m-sequence {b,} = Tr.(a"), (5.143) and decimation by J of {bn} results in the sequence {f,} = Tra("). (5.144) When J is relatively prime to the period qd - 1 of {bn}, then is another primitive element of GF(q) and, thus, {fn} is an m-sequence with the mini- mum polynomial mo(z) for its characteristic polynomial. The results on dec- imations of m-sequences can now be summarized. THEOREM 5.8. Let {bn} denote an m-sequence over GF(q) with linear span L. Then: (a) A decimation {b.j}} of {bn} is an m-sequence if and only if J is relatively prime to qL - 1. (b) Two sequences {b.j}} and {bkn} produced by decimati