Further results on Goppa codes and their applications to constructing efficient binary codes
Further results on Goppa codes and their applications to constructing efficient binary codes
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Goppa 码的进一步结果及其在构建高效二进制码中的应用
DOI：
10.1109/tit.1976.1055610
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发表时间：
1976
期刊：
影响因子：
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通讯作者：
T. Namekawa
中科院分区：
文献类型：
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作者：
Y. Sugiyama;M. Kasahara;S. Hirasawa;T. Namekawa
结果表明，具有 Goppa 多项式 \{g(z)\}^{q} 的 Goppa 码具有参数：长度 n \leq q^{m} - s_{o} 、校验符号数量 n - k \leq m (q - 1) (\deg g) 和最小距离 d \geq q (\deg g) + 1 ，其中 q 是素数幂，m 是整数，g(z ) 是任意多项式GF(q^{m}) ，g(z) 属于 GF(q^{m}) 的根数也是如此。还表明，所有长度为 n \leq 2^{m} - s_{o} 的二进制 Goppa 码都满足关系 n - k \leq m (d - 1)/2 。构造了一类新的二进制代码，其中 n \leq 2^{ m} + ms _{0}、n - k \leq m (\deg g) + s_{0} 和 d \leq 2(\deg g) + 1 ，以及另一类参数略有不同的二进制代码。其中一些代码被证明优于以前已知的最佳代码。最后给出了利用欧几里得算法构造的代码的译码算法。
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.