## ABSTRACT

Tj maps F2n to F2 and T1 is the usual trace function. For n = 2 and j = 3, we define T3(β) = 0 for all β ∈ F4. For an integer r with 1 ≤ r ≤ n, define F (n, t1, t2, . . . , tr) to be the number of elements β ∈ F2n with Tj(β) = tj for j = 1, . . . , r and let (I2(n, t1, t2, . . . , tr) =) I(n, t1, t2, . . . , tr) be the number of monic irreducible polynomials f(x) over F2 of degree n with coefficient of xn−j = tj for j = 1, . . . , r.