a larger collection of difference sets possessing the same parameters as displayed in (5.182). More precisely, when the inte- ger M in (5.182) is composite, i.e., M=JK, (5.184) then some of the difference sets created by the GMW construction do not correspond to Singer sets. The binary sequences, called GMW sequences, derived from GMW difference sets, have correlation properties identical to those of m-sequences, but possess larger linear span than m-sequences. Some of the difference sets in the GMW construction have incidence vec- tors whose elements, considered h ere as being in GF(2), can be specified as (5.185) where a is a primitive element of GF(2M), and r is any integer relatively prime to 2' - 1, r in - the range 0<r<2'- 1. When r=1, then (5.185) reduces by (5.125) to the trace representation of an m-sequence. Notice also that the interior trace function in (5.185) is an m-sequence over GF(2), and has the following property. LEMMA 5.1. Let an m-sequence over GF(q), q 2, be defined by b'n = Tra (a"), (5.186) where a is a primitive element of GF(q R), and let (5.187) Then every segment of T consecutive symbols from {b'n} contains exactly Proof. The field element T has order q - 1 and, hence, belongs to the ground field GF(q). Therefore, by the linearity property (5.123) of the trace function, it follows that for any integer k, (5.188) Since a-kT is not zero, it follows that when one trace in (5.188) is zero, SO is the other; thus, the zero locations in {b'n} are subject to a T periodicity. Consequently, every segment of T symbols from {b'n} contains the same num- ber of zeros, that number being easily evaluated from the fact that q K-1 1 zeroes occur in one period, qK - 1 symbols, of {b'n}. It will now be verified directly that the GMW sequences of (5.185) have the same autocorrelation properties as m-sequences. Pseudonoise Generators THEOREM 5.12. Let {an} be a GMW sequence whose elements are given by (5.189) where a is a primitive element of GF(2M), and r, 0 <r< 2 - 1, is relatively prime to 2 - Then the 