An $m\times (k+r)$ binary maximum distance separable (MDS) array code contains $k$ information columns and $r$ parity columns with each entry being a bit, where any $k$ out of $k+r$ columns can recover the $k$ information columns. When there is a failed column, it is critical to minimize the repair bandwidth that is the total number of bits downloaded from $d$ out of $k+r-1$ surviving columns in repairing the failed column. In this article, we first propose two explicit constructions of binary MDS array codes that have asymptotically optimal repair bandwidth for any information column, where $r\geq 2$ and $d=k+r-1$ for the first construction, and $r\geq 4$ is an even number and $d=k+\frac {r}{2}-1$ for the second construction. By applying a generic transformation for the proposed two classes of binary MDS array codes, we then obtain two classes of new binary MDS array codes that also have optimal repair bandwidth for any parity column and asymptotically optimal repair bandwidth for any information column.

中文翻译：

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任意单列渐近最优修复的两类二元MDS数组码

一个 $m\times (k+r)$ 二进制最大距离可分 (MDS) 数组代码包含 $千$ 信息栏和 $r$ 每个条目都有一点的奇偶校验列，其中任何 $千$ 在......之外 $k+r$ 列可以恢复 $千$ 信息栏。当有一个失败的列时，最小化修复带宽是至关重要的，即从下载的总位数 $d$ 在......之外 $k+r-1$ 修复故障列的幸存列。在本文中，我们首先提出了二进制 MDS 阵列代码的两种显式构造，它们对任何信息列都具有渐近最优的修复带宽，其中 $r\geq 2$ 和 $d=k+r-1$ 对于第一次建设，和 $r\geq 4$ 是偶数并且 $d=k+\frac {r}{2}-1$ 为第二次施工。通过对所提出的两类二进制 MDS 阵列代码应用通用变换，我们然后获得两类新的二进制 MDS 阵列代码，它们对于任何奇偶校验列也具有最佳修复带宽和对任何信息列具有渐近最优修复带宽。