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
(n,k,d) = (2^m + 1, 2^m - 2m, 6)
are isomorphic to the reversible cyclic codes with check polynomial
(x + 1)f(x)
, where
f(x)
is an irreducible polynomial of period
2^m + 1
.