On quasi-perfect property of double-error-correcting Goppa codes and their complete decoding

In this paper, it is proved that the binary Goppa codes with L = GF(2^m) and G(z) = z² + z + β are quasi-perfect when m is odd and are nearly-quasi-perfect when m is even. In particular, it is shown that for any syndrome, except the case where m is even and the syndrome terms are s₁ = 0 and s₃ = 1, the corresponding coset is of weight ⩽3. For the exceptional case, it is shown that the corresponding coset is of weight 4. The results thus complement to a large extent those previously reported by Moreno. Furthermore, the proofs given are constructive and offer a method of complete decoding of such Goppa codes.