Explicit Factorization Of A Class Of Polynomials Over Finite Field And Their Applications

Let Fq denote a finite field of order q,where q is the power of odd prime.In this thesis,we use some results on the factorizations of polynomial to get the irreducible factorization of a class of polynomials.Moreover,we obtain the check polynomial,dimensions,and the minimum distance of a class of irreducible negacyclic codes.Detailed works and results are as follows:(1)We give the explicit factorization of xN±a in Fq,where N = 2m pn.For any a∈Fq,there exsit β∈Fq satisfying a =βM,where M.,m,n are positive integers,p is odd prime divisor of q-1,and p≠2,The results show that the irreducible factors of xN ± a in Fq is either binomials or trinomials.(2)On the basis of(1),we give another explicit factorization of xN +1 in Fq and all primitive idempotents in the ring Fq][x]/<xN+1>,where q-1 is divisible by the prime factors of N Moreover,we obtain the check polynomial,dimensions,and the minimum distance of all irreducible negacyclic codes of length N over Fq.