Cyclotomic Sequences And Primitive Idempotents

In this thesis,we use Whiteman’s generalized cyclotomic sets of order two and classical cyclotomic sets to construct Binary pseudo-random sequence and minimal cyclic codes, investigate their perperties.Detailed results are as follows:In Chapter two,we use Whiteman’s generalized cyclotomic sets of order two to construct a binary sequence of length2pq,where p,q are different odd primes and satisfies gcd(p一1,q一1)=2. Then we can discuss when p,q satisfy certain conditions the sequences would have good linear complexity.In Chapter three,we use the classic cyclotomic to get all the2mn+m+n+1l-cyclotomic cosets,and all the2mn+m+n+1primitive idempotents in the ring Rpnqm=GF(l)[x]/(xpnqm-1), where p,q are pairwise distinct odd primes,primes lis a primitive root modulo both pn andqm, and gcd(2/φ(p2),2/φ(q2))=1,where φ is the Euler’s phi function.The minimum distance,the generator polynomial and the dimension of these minimal cyclic codes of length pnqm over GF(l)are also investigated.In Chapter four,all the6n+4l-cyclotomic cosets and6n+4primitive idempotents in the ring R2pnq=GF(l)[x]/(x2Pnq-1)are determined,where p,q,l are pairwise distinct odd primes,lis a primitive root modulo both pn and q,and gcd(2/p(p-1),2/q-1)=1.We also investigate the minimum distance,the generator polynomial and the dimension of these minimal cyclic codes of length2pnq over GF(l).