Extended double-error-correcting binary Goppa codes are cyclic (Corresp.)

The class of codes introduced by Goppa [1]-[3] includes the BCH codes as a proper subset. It also includes a large subset of asymptotically good codes, each of which has an algebraic decoding algorithm for correcting some smaller number of errors. In Section 7 of [1], Goppa gives necessary and sufficient conditions for his codes to be isomorphic to cyclic codes under a certain correspondence. In this correspondence, we exhibit another correspondence which reveals that certain other Goppa codes (including the example of Goppa\´s Section 6) become cyclic when extended by an overall parity check. In particular, the extended Goppa codes with are isomorphic to the reversible cyclic codes with check polynomial , where is an irreducible polynomial of period .