论文标题
部分可观测时空混沌系统的无模型预测
On MDS Condition and Erased Lines Recovery of Generalized Expanded-Blaum-Roth Codes and Generalized Blaum-Roth Codes
论文作者
论文摘要
与本地可回收的代码（LRC）相比，广义扩展的蓝光 - 罗斯（GEBR）代码[1]是为单符号故障，多列故障和多排故障的大规模分布式存储系统而设计的，与本地可回收的代码（LRC）相比，具有更大的可恢复性。 GEBR代码将$α\ times k $信息阵列编码为$pτ\ times（k+r）$阵列，使得斜率$ i $的线与$ 0 \ leq i \ leq i \ leq i \ leq r-1 $甚至具有平价，并且每列包含$pτ-α$ partial parity sartity prients prients符号，其中$ p $是奇怪的prime和$ k+r r+r r \ r \ leq pest p p p p $ \ leq pest p p p p $ GEBR代码为$（n，k）$可恢复的必要条件（即，$ n = k+r $列中的任何$ k $都可以检索所有信息符号），以$α=（p-1）τ$进行[2]中。但是，当$α<（p-1）τ$时，$（n，k）$可回收条件是未知的。在本文中，我们介绍了$α<（p-1）τ$的$（n，k）$可回收条件。此外，我们提供了一个足够的条件，可以使GEBR代码恢复任何坡度$ i $（$ 0 \ leq i \ leqpτ-1 $）的删除行，当$τ$是$ p $的电源时，任何参数$ r $。此外，我们介绍将$α\ times k $信息数组编码为$α\ times（k+r）$阵列的广义蓝光（GBR）代码的构建。我们表明，GBR码具有与GEBR代码的$（N，K）$可回收条件相同的MDS条件，并且我们还为GBR代码提供了足够的条件，可以恢复任何坡度$ i $的删除行（$ 0 \ leq i \ leq i \ leq leqα-1$）。
Generalized Expanded-Blaum-Roth (GEBR) codes [1] are designed for large-scale distributed storage systems that have larger recoverability for single-symbol failures, multi-column failures and multi-row failures, compared with locally recoverable codes (LRC). GEBR codes encode an $α\times k$ information array into a $pτ\times (k+r)$ array such that lines of slope $i$ with $0\leq i\leq r-1$ have even parity and each column contains $pτ-α$ local parity symbols, where $p$ is an odd prime and $k+r\leq pτ$. Necessary and sufficient conditions for GEBR codes to be $(n,k)$ recoverable (i.e., any $k$ out of $n=k+r$ columns can retrieve all information symbols) are given in [2] for $α=(p-1)τ$. However, the $(n,k)$ recoverable condition of GEBR codes is unknown when $α<(p-1)τ$. In this paper, we present the $(n,k)$ recoverable condition for GEBR codes for $α< (p-1)τ$. In addition, we present a sufficient condition for enabling GEBR codes to recover some erased lines of any slope $i$ ($0\leq i\leq pτ-1$) for any parameter $r$ when $τ$ is a power of $p$. Moreover, we present the construction of Generalized Blaum-Roth (GBR) codes that encode an $α\times k$ information array into an $α\times (k+r)$ array. We show that GBR codes share the same MDS condition as the $(n,k)$ recoverable condition of GEBR codes, and we also present a sufficient condition for GBR codes to recover some erased lines of any slope $i$ ($0\leq i\leq α-1$).
