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We consider asymptotically good nonlinear codes recently introduced by Xing (2003). The original definition of these codes relies on a nonconstructive averaging argument. In this paper, it is first shown that in some cases, the codes can be constructed without using any averaging arguments. We then introduce an alternative construction of the codes, based on the union of a geometric Goppa code and its cosets. In some cases, the problem of explicitly describing the codes reduces to the problem of explicitly describing certain n elements of the relevant function field, where n is the code length. Moreover, the number of finite-field operations required to construct these n elements after the construction of the generator matrix of the geometric Goppa code is of the order of n3.