Primitive Idempotents Of Irreducible Cyclic Codes

Cyclic codes is a class of the most widely used error-correcting codes at present,which can be generated by idempotents.The generator idempotents of irreducible cyclic codes are called primitive idempotents.In many applications,the use of idempotents as generator polynomials of cyclic codes is advantageous to their research,and a lot of papers investigate relevant results on idempotents.This thesis mainly uses some results on irreducible factorizations of polynomials over finite fields and character matrix of finite groups method to give all primitive idempotents in two classes of rings.Detailed works are as follow:(1)Let Fq=FPn be a finite extension of Fp,where n is a positive integer with gcd(n,p)=1.If a? Fp*,then we give the irreducible factorizations of affine trinomials xq-x-a over Fqn.By character matrix method,we obtain expression forms of q/p primitive idempotents explicitly inthe ring Fq[x]/<xq-x-a>.(2)Let Fq be a finite extension of Fp and n is a positive integer with gcd(n,p)=1.If theproduct of all prime divisors of the positive integer n divide q-1,then Marinez and Oliveira explicitly factorized xn-1 over Fq in the two cases:either q(?)3(mod 4)or 8+n;q?3(mod 4)and 8|n.Based on this conclusion and character matrix method,we give the explicit expression forms of all primitive idempotents of irreducible cyclic codes of length n over Fq,and we also determine the minimum Hamming distances of irreducible cyclic codes of length n in the case of whose parity-check polynomials are binomials.