The minimum Hamming distances of the irreducible cyclic codes of length l(1)(m1) l(2)(m2) ... l(r)(mr)

Let F-q be a finite field with q elements and n = l(1)(m1) l(2)(m2) ... l(r)(mr), m(1) >= 1, m(2) >= 1, ..., m(r) >= 1, where l(1), l(2), ...., l(r) are distinct odd primes and l(1) l(2) .... l(r) vertical bar q - 1. In this paper, we give all irreducible factors of x(l1m1l2m2) (... lrmr) - 1 over F-q, all primitive idempotents in the ring F-q[x]/< x(l1m1l2m2) (... lrmr) - 1 >. Furthermore, we obtain the dimensions and the minimum Hamming distances of all irreducible cyclic codes of length l(1)(m1) l(2)(m2) ... l(r)(mr) over F-q.