A maximum distance separable (MDS) array code is composed of $m\times (k+r)$ arrays such that any $k$ out of $k+r$ columns suffice to retrieve all the information symbols. Expanded-Blaum-Roth (EBR) codes and Expanded-Independent-Parity (EIP) codes are two classes of MDS array codes that can repair any one symbol in a column by locally accessing some other symbols within the column, where the number of symbols $m$ in a column is a prime number. By generalizing the constructions of EBR and EIP codes, we propose new MDS array codes, such that any one symbol can be locally recovered and the number of symbols in a column can be not only a prime number but also a power of an odd prime number. Also, we present an efficient encoding/decoding method for the proposed generalized EBR (GEBR) and generalized EIP (GEIP) codes based on the LU factorization of a Vandermonde matrix. We show that the proposed decoding method has less computational complexity than existing methods. Furthermore, we show that the proposed GEBR codes have both a larger minimum symbol distance and a larger recovery ability of erased lines for some parameters when compared to EBR codes. We show that EBR codes can recover any $r$ erased lines of a slope for any parameter $r$, which was an open problem in [2].

翻译：最大距离分解( MDS) 阵列代码由 $m\ time (k+r) 阵列组成, 如此一来, 美元+r 列中的任何美元都足以收回所有信息符号。 扩展- Blaum- Roth (EBR) 代码和扩展- 独立- Paity (EIP) 代码是 扩展- Blaum- Roth (EBR) 代码的两大类 MDS 阵列代码, 这些代码可以通过本地访问一列中某些其它符号来修复一列中的任何单个符号, 其中一列中的符号数数是质数。 我们通过概括 EBR 和 EIP 的构建, 我们提出了新的阵列代码, 其中的计算复杂性比现有方法要小一些。 此外, 我们为回收的 EBR 提供了一种最小值的编码。 我们为 EBR 提供了一种最小的代号, 当我们为 EBR 的代号 显示一个最小的代号时, 我们为 EBR 的恢复 。