Codes on the Klein quartic, ideals, and decoding
Codes on the Klein quartic, ideals, and decoding
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克莱因四次方程的代码、理想和解码
DOI：
10.1109/tit.1987.1057365
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发表时间：
1987
期刊：
影响因子：
--
通讯作者：
J. Hansen
中科院分区：
文献类型：
--
作者：
J. Hansen
在域 GF (2^{3}) 上构建具有特定对称性且与最小距离相比具有较大速率的代码序列。在该序列中，例如存在具有最小距离9的长度21和尺寸10的代码，以及具有最小距离3的长度21和尺寸16的代码。这些代码是使用 Goppa 建立的编码理论和有限域上的代数曲线之间的字典从代数几何构造的。本工作中使用的曲线是克莱因四次曲线。该曲线具有 Serre 对 Hasse-Weil 界的改进所允许的 GF (2^{3}) 上有理点的最大数量，这与低亏格一起说明了良好的代码参数。克莱因四次方程具有 21 阶弗罗贝尼乌斯群 G，作为一组自同构，解释了码的特殊对称性。事实上，这些代码在群代数 GF (2^{3})[G] 中被给予了替代描述作为左理想。该描述允许轻松解码。例如，在长度为21、尺寸为16且最小距离为3的单个纠错码的情况下。解码是通过与群代数中的幂等相乘获得的。
A sequence of codes with particular symmetries and with large rates compared to their minimal distances is constructed over the field GF (2^{3}) . In the sequence there is, for instance, a code of length 21 and dimension 10 with minimal distance 9 , and a code of length 21 and dimension 16 with minimal distance 3 . The codes are constructed from algebraic geometry using the dictionary between coding theory and algebraic curves over finite fields established by Goppa. The curve used in the present work is the Klein quartic. This curve has the maximal number of rational points over GF (2^{3}) allowed by Serre's improvement of the Hasse-Weil bound, which, together with the low genus, accounts for the good code parameters. The Klein quartic has the Frobenius group G of order 21 acting as a group of automorphisms which accounts for the particular symmetries of the codes. In fact, the codes are given alternative descriptions as left ideals in the group-algebra GF (2^{3})[G] . This description allows for easy decoding. For instance, in the case of the single error correcting code of length 21 and dimension 16 with minimal distance 3 . decoding is obtained by multiplication with an idempotent in the group algebra.