Further results on Goppa codes and their applications to constructing efficient binary codes

It is shown that Goppa codes with Goppa polynomial\{g(z)\}qhave the parameters: lengthn ≤ qᵐ - sₒ, number of check symbolsn - k ≤ m (q - 1) ( g), and minimum distanced ≥ q ( g) + 1, whereqis a prime power,mis an integer,g(z )is an arbitrary polynomial overGF(qᵐ), and so is the number of roots ofg(z)which belong toGF(qᵐ). It is also shown that all binary Goppa codes of lengthn ≤ 2ᵐ - sₒsatisfy the relationn - k ≤ m (d - 1)/2. A new class of binary codes withn ≤ 2ᵐ + ms ₀, n - k ≤ m ( g) + s₀, andd ≤ 2( g) + 1is constructed, as well as another class of binary codes with slightly different parameters. Some of those codes are proved superior to the best codes previously known. Finally, a decoding algorithm is given for the codes constructed which uses Euclid's algorithm.