arXiv 1503.090922015-03cs.ITmath.IT

从随机错误中有效地译码Reed-Muller码

Efficiently decoding Reed-Muller codes from random errors

Ramprasad Saptharishi, Amir Shpilka, Ben Lee Volk

备注：: 18 pages, 2 figures

摘要：

Reed-Muller码通过在$\{0,1\}^m$中的所有点上对$r$次数的$m$变量多项式进行求值来对$m$变量多项式进行编码。我们用$RM(m,r)$表示该代码。$RM(m,r)$的最小距离是$2^{m-r}$，因此在最坏的情况下，它不能纠正超过一半的错误。对于随机误差，人们可能希望得到更好的结果。在这项工作中，我们给出了一个有效的算法（在块长度$n=2^m$）解码的Reed-Muller码的随机错误远远超过最小距离。具体地，对于低速率码（度为$r=o(\sqrt{m})$），我们可以以高概率校正$(1/2-o(1))n$错误的随机集合。对于高速率码（对于$r=o(\sqrt{m/\log m})$为$m-r$度），我们可以粗略地校正$m^{r/2}$误差。更一般地，对于任何整数$r$，我们的算法可以纠正$RM(m,m-(2r+2))$中的任何错误模式，对于该错误模式，可以在$RM(m,m-(r+1))$中纠正相同的擦除模式。上述结果是通过应用Abbe，Shpilka和Wigderson（STOC，2015），Kumar和Pfister（2015）以及Kudekar等人（2015）关于Reed-Muller码纠正随机擦除能力的最新结果获得的。该算法是基于求解一组仔细定义的线性方程组，因此它是显着不同于其他算法解码Reed-Muller码的递归结构的基础上的代码。它可以被看作是对Abbe等人的一个结果的更明确的证明，该结果表明了从纠正擦除到纠正错误的减少，并且它也与著名的Berlekamp-Welch算法解码Reed-Solomon码有一些相似之处。

英文摘要：

Reed-Muller codes encode an $m$-variate polynomial of degree $r$ by evaluating it on all points in $\{0,1\}^m$. We denote this code by $RM(m,r)$. The minimal distance of $RM(m,r)$ is $2^{m-r}$ and so it cannot correct more than half that number of errors in the worst case. For random errors one may hope for a better result. In this work we give an efficient algorithm (in the block length $n=2^m$) for decoding random errors in Reed-Muller codes far beyond the minimal distance. Specifically, for low rate codes (of degree $r=o(\sqrt{m})$) we can correct a random set of $(1/2-o(1))n$ errors with high probability. For high rate codes (of degree $m-r$ for $r=o(\sqrt{m/\log m})$), we can correct roughly $m^{r/2}$ errors. More generally, for any integer $r$, our algorithm can correct any error pattern in $RM(m,m-(2r+2))$ for which the same erasure pattern can be corrected in $RM(m,m-(r+1))$. The results above are obtained by applying recent results of Abbe, Shpilka and Wigderson (STOC, 2015), Kumar and Pfister (2015) and Kudekar et al. (2015) regarding the ability of Reed-Muller codes to correct random erasures. The algorithm is based on solving a carefully defined set of linear equations and thus it is significantly different than other algorithms for decoding Reed-Muller codes that are based on the recursive structure of the code. It can be seen as a more explicit proof of a result of Abbe et al. that shows a reduction from correcting erasures to correcting errors, and it also bares some similarities with the famous Berlekamp-Welch algorithm for decoding Reed-Solomon codes.

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英文摘要：

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