The study of Goppa codes (Goppa 1970) is important for at least the following reasons.
(a)Goppa codes generalize the narrow sense BCH codes.
(b)Their class contains arbitrarily long q-ary codes the dmin/n of which strictly exceeds the asymptotic Gilbert bound for all q ≥ 49 and for every rate R in a certain interval depending on q.
(c)They can be efficiently decoded up to their designed distance.
(d)They have been proposed for a public-key cryptosystem (McEliece 1978).
Before Goppa codes are formally defined some of the classic theory of BCH codes is reviewed, however, cast in a light appropriate to what is needed later.
LEMMA 9.1 The n-tuple c = (c0, c1, ..., cn − 1) ∈ GF(q)n is a codeword of the narrow ...