The minimum Hamming distances of irreducible cyclic codes

Let FqFq be a finite field with q   elements and n=l1m1l2m2, m1≥1m1≥1, m2≥1m2≥1, where l1l1, l2l2 are distinct primes and l1l2|q−1l1l2|q−1. In this paper, we give all irreducible factors of xl1m1l2m2−1 over FqFq and all primitive idempotents in a ring Fq[x]/〈xl1m1l2m2−1〉. Furthermore, we obtain the dimensions and the minimum Hamming distances of all irreducible cyclic codes of length l1m1l2m2 over FqFq.