مرکز منطقه ای اطلاع رساني علوم و فناوري - Further results on Goppa codes and their applications to constructing efficient binary codes

DocumentCode :

925722

Title :

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

Author :

Sugiyama, Yasuo ; Kasahara, Masao ; Hirasawa, Shigeichi ; Namekawa, Toshihiko

Volume :

Issue :

fYear :

1976

fDate :

9/1/1976 12:00:00 AM

Firstpage :

518

Lastpage :

526

Abstract :

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 power, $m$ 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 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.