New identities relating wild Goppa codes

For a given support LεF_qmⁿ and a polynomial gεF_qmⁿ [x] with no roots in F_qm ⁿ, we prove equality between the q-ary Goppa codes ^Γq(L,N(g))=^Γq(L,N(g)/g) where N(g) denotes the norm of g, that is gqm-¹+⋯+q+1. In particular, for m=2, that is, for a quadratic extension, we get ^Γq(L,^gq)= ^Γq(L,gq+¹). If g has roots in F_qm ⁿ, then we do not necessarily have equality and we prove that the difference of the dimensions of the two codes is bounded above by the number of distinct roots of g in ^Fqm. These identities provide numerous code equivalences and improved designed parameters for some families of classical Goppa codes.