On period polynomials of degree 2m and weight distributions of certain irreducible cyclic codes

We explicitly determine the values of reduced cyclotomic periods of order 2m, m≥4, for finite fields of characteristic p≡3 or 5(mod8). These evaluations are applied to obtain explicit factorizations of the corresponding reduced period polynomials. As another application, the weight distributions of certain irreducible cyclic codes are described.