A class of minimal cyclic codes over finite fields

Let FqFq be a finite field of odd order qq and n=2ap1a1p2a2, where a,a1,a2a,a1,a2 are positive integers, p1,p2p1,p2 are distinct odd primes and 4p1p2|q−14p1p2|q−1. In this paper, we study the irreducible factorization of xn−1xn−1 over FqFq and all primitive idempotents in the ring Fq[x]∕〈xn−1〉Fq[x]∕〈xn−1〉.Moreover, we obtain the dimensions and the minimum Hamming distances of all irreducible cyclic codes of length nn over FqFq.