Abstract
In this chapter it is assumed that the reader is familiar with algebraic coding theory. A reader without this background can freely skip this chapter and continue with Chapter 11. From [Mac77] we recall the following facts about Goppa codes. With each irreducible polynomial of degree tover GF(2m) corresponds a binary, irreducible Goppa code of length n = 2m, dimension k ≥ n - tm and minimum distance d ≥ 2t + 1. A fast decoding algorithm with running time nt, exists [Pat75]. There are about 2mt/t (see Corollary B.23) irreducible polynomials of degree t over GF(2m). So a random polynomial of degree t over GF(2m) will be irreducible with probability 1/t. Since there is a fast algorithm for testing irreducibility (see [Ber68, Ch.6] or [Rab80]), one can find an irreducible polynomial of degree t over GF(2m), just like in Algorithm 9.5, by repeated guessing and testing.
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