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Elements of Algebraic Coding Systems by Valdemar Cardoso da Rocha, Jr.

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Chapter 9

GOPPA CODES

9.1 Introduction

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 ...

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