, and the corresponding LFSR supports only two cyclically distinct state sequences, namely the per- petual zero-state sequence and the state sequence of {bn}. Property T-4 indi- cates that Tr(a") must be non-zero for some n; hence, the state sequence corresponding to the trace sequence cannot be the zero-state sequence. Therefore, the trace sequence or some shift of it must be the m-sequence. That is, {Tr(am+")} { {b,} (5.131) for some m, and setting y = am completes the proof. 5.5.5 A Correlation Computation To illustrate the use of trace representations in the study of periodic corre- lation properties, consider the calculation of the periodic correlation of {an}, a roots-of-unity m-sequence derived from an m-sequence {bn} over GF(q) via (5.80). Letma(z) denote the degree d primitive characteristic polynomial of {bn}. Then the periodic correlation function of {an}, defined in (5.85), can be rewritten in terms of the trace function by using Corollary 5.2 and sim- plifying with Property T-3. (5.132) where y(a - 1). (5.133) Galois Field Connections Since a is primitive, an takes on the values of all the non-zero elements in GF(q) as n goes from 1 to N, N being the sequence period. Adding the zero element to this list and noting that its trace is zero, yields (5.134) B EGF(qd) As B varies over GF(q), SO does SB unless 8 is zero, and applying Property T-4, gives (5.135) The condition 8 = 0 occurs if, and only if, = 1, and since a is primitive, = 0iff = 0 modq - - 1. (5.136) While this has been a somewhat more involved proof of randomness Property R-5 than that given in Section 5.4.4, the trace function methodol- ogy employed here readily generalizes to several more complicated full- period correlation calculations. 5.5.6 Decimations of Sequences Let {bn} be an arbitrary sequence of period N, and consider the sequence {cn} defined by Cn=bJn, for all N. (5.137) The sequence {cn} is said to be the decimation by J of the sequence {bn}. If J divides N, then {cn} has period NIJ. On the other hand, if J and N are rela- tiv