 84672 120032 356962 276480 1296000 1719900 4202496 4741632 18407808 of sequences, which are not cyclically equivalent, are termed cyclically dis- tinct. Since m-sequences are cyclically distinct if and only if their linear recur- sions differ, the number of cyclically distinct m-sequences over GF(q) with linear span L is equal to the number Np(L) of primitive polynomials of degree L over GF(q), and is given by (5.57) where the J prime numbers pi,i = J are determined from the prime decomposition of q - - 1,i.e., q' - 1 = II p (5.58) where = 1,...,J , are positive integers. Table 5.6 illustrates the expo- nential nature of Np(L) for binary m-sequences. 5.4 STATISTICAL PROPERTIES OF M-SEQUENCES Primitive polynomials provide LFSR connections for m-sequence genera- tors which are virtually as efficient as counters in providing state sequences with large periods, the LFSR's state sequence being shorter by one than the counter's period for the same memory size. However, the binary counter did not provide a good basis for a PN generator because the individual mem- ory cells of the counter produced simple "square wave" sequences, thereby requiring significant amounts of logic to produce sequences with the appear- ance of randomness. As will now be demonstrated, each memory cell of a Statistical Properties of M-Sequences Fibonacci m-sequence generator produces a sequence with statistical prop- erties far superior, for PN applications, to a counter's cell output. 5.4.1 Event Counts Since m-sequences over GF(q) with linear span L have period q1 - 1 and since the all-zeros sequence of self-perpetuating in a linear generator, it immediately follows that an m-sequence generator must cycle through the q1 - 1 non-zero state L-tuples. In particular, a Fibonacci generator cycles through the state L-tuples (5.59) bn+L-1 which correspond to L-symbol segments of the m-sequence {bn} (see (5.30)- (5.32)). This viewpoint leads to verification of the following random-appear- ance properties: Property R-1. The number Nb of times that a no