Further results on Goppa codes and their applications to constructing efficient binary codes
Yasuo Sugiyama, M. Kasahara, Shigeichi Hirasawa, T. Namekawa · IEEE Transactions on Information Theory · 1976
It is shown that Goppa codes with Goppa polynomial\{g(z)\}^{q}have the parameters: lengthn \leq q^{m} - s_{o}, number of check symbolsn - k \leq m (q - 1) (\deg g), and minimum distanced \geq q (\deg g) + 1, whereqis a prime power,mis an integer,g(z )is an arbitrary polynomial overGF(q^{m}), and so is the number of roots ofg(z)which belong toGF(q^{m}). It is also shown that all binary Goppa codes of lengthn \leq 2^{m} - s_{o}satisfy the relationn - k \leq m (d - 1)/2. A new class of binary codes withn \leq 2^{ m} + ms _{0}, n - k \leq m (\deg g) + s_{0}, andd \leq 2(\deg 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.