New Identities Relating Wild Goppa Codes

Original · EN

For a given support L ∈ Fqₘⁿ and a polynomial g∈ Fqₘ[x] with no roots in Fqₘ, 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 Fqₘ, 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.