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On the Shape of the General Error Locator Polynomial for Cyclic Codes

机译:关于循环码的一般错误定位多项式的形状

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摘要

General error locator polynomials were introduced in 2005 as an alternative decoding for cyclic codes. We present now a conjecture on their sparsity which would imply polynomial-time decoding for all cyclic codes. A general result on the explicit form of the general error locator polynomial for all cyclic codes is given, along with several results for specific code families, providing evidence to our conjecture. From these, a theoretical justification of the sparsity of general error locator polynomials is obtained for all binary cyclic codes with t = 2 and n105, as well as for t=3 and n63, except for some cases where the conjectured sparsity is proved by a computer check. Moreover, we summarize all related results, previously published, and we show how they provide further evidence to our conjecture. Finally, we discuss the link between our conjecture and the complexity of bounded-distance decoding of cyclic codes.
机译:通用错误定位器多项式于2005年引入,作为循环码的另一种解码方法。现在我们对它们的稀疏性提出一个推测,这暗示着对所有循环码进行多项式时间解码。给出了针对所有循环码的一般错误定位符多项式的显式形式的一般结果,以及针对特定代码族的若干结果,为我们的猜想提供了证据。由此,对于t <= 2和n <105的所有二进制循环码,以及对于t = 3和n <63的所有二进制循环码,获得了一般错误定位符多项式的稀疏性的理论证明,除了一些推测的情况稀疏性通过计算机检查来证明。此外,我们总结了先前发布的所有相关结果,并展示了它们如何为我们的猜想提供进一步的证据。最后,我们讨论了猜想与循环码有界解码的复杂度之间的联系。

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