A class of binary cyclic codes with optimal parameters

For positive integers k >= 2 and t, let m = 2kt and alpha be a primitive element of the finite field F-2m. In this paper, we study the parameters of a class of cyclic codes C-(1,C-v) which has two zeros alpha and alpha(v) with v = 2(m)-1/2(t)+1 It is shown that C-(1,C-v) is optimal or almost optimal with respect to the sphere packing bound. Based on some results of Kloosterman sums and Gaussian periods, the weight distribution of the dual code of C-(1,C-v) is completely determined when t = 5.