Gauss Periods, Cyclotomic Sequences And Codebooks

First, We will use Whiteman’s generalized cyclotomic sets of order four and classical cyclotomic sets to construct pseudorandom sequences and codebooks, investigate their perperties; second, calculate the Gauss periods of order three and six, as an application, get the linear complexity of two3-ary sequences from cyclotomic sets; third, give a method to compute the linear complexity of a Whiteman’s generalized cyclotomic binary sequence of period pm+1qn+1, in special, give the exact linear complexity of the generalized cyclotomic sequence if gcd(p(p-1),q(q-1))=4. Results are as follows in detail:In Chapter two, we use Whiteman’s generalized cyclotomic sets of order four to construct a binary sequence, and give conditions of p, q such that the autocorrelation of the binary sequence is well.In Chapter three, we use Whiteman’s generalized cyclotomic sets of order four to construct two codebooks, and give conditions of p,q such that the cross-correlation of the codebook is nearly reach to Welch bound.In Chapter four, we use the classic cyclotomic numbers of order3and6to get the cyclotomic numbers modulo three, then, use these cyclotomic numbers to calculate the Gauss periods from cyclotomic sets of order3and6. As applications, we get the linear complexity of two3-ary sequences from cyclotomic sets.In Chapter five, we give a method to compute the linear complexity of a Whiteman’s generalized cyclotomic binary sequence of period pm+1qn+1, where p, q are odd primes and gcd(p(p-1), q(q-1))=e. As a special case, we give the exact linear complexity of the generalized cyclotomic sequence. Then, let e=4, we compute the exact linear complexity of the genenralized cyclotomic sequence. At last, let p=q=5(mod8), gcd(p(p-1), q(q-1))=4, and fix a common primitive root g of both p and q, we will determine Whiteman’s generalized cyclotomic number from pq=a2+4b2,4|b.