关于纠两个错误的二元Goppa码的准完备性及其完全译码

本文证明了位置集L=GF(2^m)、生成多项式G(z)=z²+az+b的二元Goppa码,除m为偶数并且S₁=0,S₃=a^-1外,是准完备的.当m为偶数,那末不存在重量不大于3的矢量具有伴随式S₁=0,S₃=a^-1;但至少存在一个重量为4的矢量具有伴随式S₁=0,S₃=a^-1,此外,还给出L=GF(2^m)、G(z)=z²+az+b的二元Goppa码的一个完全译码.

ON QUASI-PERFECT PROPERTY OF DOUBLE-ERROR-CORRECTING GOPPA CODES AND THEIR COMPLETE DECODING

In this paper, we prove that the binary Goppa codes with L=GF(2^m) and G(z)=z²+az+b are quasi-perfect unless m is even and S₁=0,S₃=a^-1. When m is even, there is not any vector of weight <3 having syndrome S₁=0 and S₃=a^-1 but there is at least a vector of weight 4 having syndrome S₁=0 and S₃=a^-1. At the same time we give also a complete decoding of the binary Goppa codes with L=GF(2^m) and G(z)=z²+az+b.