Goppa codes are a well-known class of linear codes with important applications in cryptography. Determining the minimum distance of Goppa codes and constructing Goppa codes with prescribed automorphism groups are both meaningful and challenging problems in coding theory. In this paper, we first study the minimum distance of binary separable Goppa codes. For the two classes $g(X)=f(X^t)$ and $g(X)=A(X)h(φ(X))$, we give criteria for attaining the designed distance and derive several infinite families whose minimum distances are determined. We then construct binary Goppa codes and their related codes with $A_4$ or $A_5$ automorphism groups. These constructions also naturally yield binary quasi-cyclic Goppa codes and their related codes. Moreover, by applying the minimum-distance criteria developed above, we determine the parameters of one class of the constructed $A_4$-invariant Goppa codes.

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