On partial maximally-recoverable and maximally-recoverable codes
On partial maximally-recoverable and maximally-recoverable codes
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关于部分最大可恢复代码和最大可恢复代码
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
P. V. Kumar
中科院分区：
文献类型：
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作者：
Balaji Srinivasan Babu;P. V. Kumar
受奇偶校验矩阵 H0 施加的局部性约束的 [n, k] 线性码 C 被称为最大可恢复 (MR) 码，如果它可以从 H0 零空间的某些 k 维子码可以恢复的任何擦除模式中恢复。本文的重点是受限于全符号局部性 r 的 MR 码。鉴于构造具有小场尺寸的 MR 代码具有挑战性，我们在两个方向上展示结果。首先，我们放宽了 MR 约束，并且仅要求除了成为最佳全符号局部性码的要求之外，该代码必须在以单个特定模式进行穿孔时产生 MDS 码，这确保每个局部码精确地在一个坐标中进行穿孔，并且没有两个局部码共享相同的穿孔坐标。我们将这些代码称为部分最大可恢复（PMR）代码。我们为高速率 PMR 代码提供了一种简单的构造，然后提供了一种需要进一步研究的通用且有前景的方法。在第二个方向上，我们提出了三种具有改进参数的 MR 码构造，主要是构造中使用的有限域的大小。
An [n, k] linear code C that is subject to locality constraints imposed by a parity check matrix H0 is said to be a maximally recoverable (MR) code if it can recover from any erasure pattern that some k-dimensional subcode of the null space of H0 can recover from. The focus in this paper is on MR codes constrained to have all-symbol locality r. Given that it is challenging to construct MR codes having small field size, we present results in two directions. In the first, we relax the MR constraint and require only that apart from the requirement of being an optimum all-symbol locality code, the code must yield an MDS code when punctured in a single, specific pattern which ensures that each local code is punctured in precisely one coordinate and that no two local codes share the same punctured coordinate. We term these codes as partially maximally recoverable (PMR) codes. We provide a simple construction for high-rate PMR codes and then provide a general, promising approach that needs further investigation. In the second direction, we present three constructions of MR codes with improved parameters, primarily the size of the finite field employed in the construction.