Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: It is shown that Goppa codes with Goppa polynomial \{g(z)\}^{q} have the parameters: length n \leq q^{m} - s_{o} , number of check symbols n - k \leq m (q - 1) (\deg g) , and minimum distance d \geq q (\deg g) + 1 , where q is a prime por, m is an integer, g(z ) is an arbitrary polynomial over GF(q^{m}) , and so is the number of roots of g(z) which belong to GF(q^{m}) . It is also shown that all binary Goppa codes of length n \leq 2^{m} - s_{o} satisfy the relation n - k \leq m (d - 1)/2 . A new class of binary codes with n \leq 2^{ m} + ms _{0}, n - k \leq m (\deg g) + s_{0} , and d \leq 2(\deg g) + 1 is constructed, as ll 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.
['Yasuo Sugiyama', 'Masao Kasahara', 'Shigeichi Hirasawa', 'Toshihiko Namekawa']
I am writing a research paper on a idea Further results on Goppa codes and their applications to constructing efficient binary codes
[{"from": "human", "value": "I am writing a research paper on a idea Further results on Goppa codes and their applications to constructing efficient binary codes"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: It is shown that Goppa codes with Goppa polynomial \\{g(z)\\}^{q} have the parameters: length n \\leq q^{m} - s_{o} , number of check symbols n - k \\leq m (q - 1) (\\deg g) , and minimum distance d \\geq q (\\deg g) + 1 , where q is a prime por, m is an integer, g(z ) is an arbitrary polynomial over GF(q^{m}) , and so is the number of roots of g(z) which belong to GF(q^{m}) . It is also shown that all binary Goppa codes of length n \\leq 2^{m} - s_{o} satisfy the relation n - k \\leq m (d - 1)/2 . A new class of binary codes with n \\leq 2^{ m} + ms _{0}, n - k \\leq m (\\deg g) + s_{0} , and d \\leq 2(\\deg g) + 1 is constructed, as ll 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."}]