In this paper, the problem of reconstruction of the feedback polynomial in a linear scrambler is studied. Our work contains two parts. In the first part, schemes to improve the performance of an existing reconstruction algorithm are proposed. Simulation results show that both the detection capability and speed of the existing algorithm are significantly improved by using our proposed schemes. In the second part, the reconstruction of linear scramblers in the presence of channel noise is investigated. We consider flipped bits due to noise as well as insertion of bits in the scrambled bit sequence. For both cases, factors which affect the performance of the reconstruction algorithm are discussed.