In this session, we will talk about another family of codes that have an efficient decoding algorithm: the Goppa codes. One limitation of the generalized Reed-Solomon codes is the fact that the length is bounded by the size of the field over which it is defined. This implies that these codes are useful when we use a large field size. In the sequence, we'll present a method to obtain a new code over small alphabets by exploiting the properties of the generalized Reed-Solomon codes. So, the idea is to construct a generalized Reed-Solomon code over a sufficiently large extension of a field and extract only those codewords that lie completely in the field. Let a be a n-tuple of elements from the field Fq^n which are all different and let b be an n-tuple of elements from the field Fq^n which are non-zeros. Then, the alternant co