论文标题
修复在子空间上评估的芦苇 - 固体代码
Repairing Reed-Solomon Codes Evaluated on Subspaces
论文作者
论文摘要
我们考虑REED的维修问题（RS）代码，对$ \ Mathbb {f} _Q $ -linear-linear subspace $ u \ subseteq \ subseteq \ mathbb {f} _ {q^m} $ d $ d $ d $的$ q $ q $ quest $ quest $ quest $ $ $ $ iSGE，以及$ Q $的Galois领域。对于$ q \ geq 3 $，我们显示了RS长度$ n = q^d $和codimension $ q^s $，$ s <d $的线性维修方案的存在，对$ u $进行了评估，其中$ n-1 $ nodes nodes transmits transmits transmits transmits transmits仅$ r $ r $ r $ r $ r $ r $ r $ r $ \ nathbbbbbb f i f f f if f iff} $ {$ r）$ {ms $ r）$ r.s n $ s $ r）$ r. s $ r. s q of。对于$ q = 2 $的情况，我们证明了类似的结果，对评估线性子空间$ u $的一些限制。我们的证明是基于概率论点的，但是结果不仅是存在结果。成功概率相当大（至少$ 1/3 $），并且有一个简单的标准来检查随机选择的线性修复方案的有效性。我们的结果将Dau-Milenkovich的构造扩展到了$ r <m-s $的范围，用于广泛的参数。
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 the case of $q\geq 3$, 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 of $q=2$, 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--Milenkovich to the range $r<m-s$, for a wide range of parameters.
