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Repairing Reed–Solomon Codes Evaluated on Subspaces
IEEE Transactions on Information Theory ( IF 2.2 ) Pub Date : 2022-05-25 , DOI: 10.1109/tit.2022.3177903
Amit Berman 1 , Sarit Buzaglo 1 , Avner Dor 1 , Yaron Shany 1 , Itzhak Tamo 2
Affiliation  

We consider the repair problem for Reed–Solomon (RS) codes, evaluated on an $\mathbb {F}_{q}$ -linear subspace $U\subseteq \mathbb {F}_{q^{m}} $ of dimension $d$ , where $q$ is a prime power, $m$ is a positive integer, and $\mathbb {F}_{q}$ is the Galois field of size $q$ . For $q>2$ , we show the existence of a linear repair scheme for the RS code of length $n=q^{d}$ and codimension $q^{s}$ , $s < d$ , evaluated on $U$ , in which each of the $n-1$ surviving nodes transmits only $r$ symbols of $\mathbb {F}_{q}$ , provided that $ms\geq d(m-r)$ . For the case $q=2$ , we prove a similar result, with some restrictions on the evaluation linear subspace $U$ . Our proof is based on a probabilistic argument, however the result is not merely an existence result; the success probability is fairly large (at least $1/3$ ) and there is a simple criterion for checking the validity of the randomly chosen linear repair scheme. Our result extend the construction of Dau–Milenkovic to the range $r < m-s$ , for a wide range of parameters.

中文翻译:

修复在子空间上求值的 Reed-Solomon 码

我们考虑 Reed-Solomon (RS) 码的修复问题,在 $\mathbb {F}_{q}$ -线性子空间 $U\subseteq \mathbb {F}_{q^{m}} $维度的 $d$ , 在哪里 $q$是主力, $m$是一个正整数,并且 $\mathbb {F}_{q}$是大小的伽罗瓦域 $q$ . 为了 $q>2$ ,我们展示了长度为 RS 码的线性修复方案的存在 $n=q^{d}$和维度 $q^{s}$ , $s < d$ , 评估在 美元 ,其中每个 $n-1$幸存的节点只传输 $r$的符号 $\mathbb {F}_{q}$ , 前提是 $ms\geq d(mr)$ . 对于案件 $q=2$ ,我们证明了类似的结果,但对评估线性子空间有一些限制 美元 . 我们的证明是基于概率论的,然而结果不仅仅是存在的结果;成功概率相当大(至少 $1/3$ ) 并且有一个简单的标准来检查随机选择的线性修复方案的有效性。我们的结果将 Dau-Milenkovic 的构造扩展到 $r < 毫秒$ , 适用于各种参数。
更新日期:2022-05-25
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