Decoding and repair schemes are proposed for shift-exclusive-or (shift-XOR) product-matrix (PM) regenerating codes, which outperform the existing schemes in terms of both communication and computation costs. In particular, for the shift-XOR minimum bandwidth regenerating (MBR) codes, our decoding and repair schemes have the optimal transmission bandwidth and can be implemented in-place without extra storage space for intermediate XOR results. Technically, our schemes involve an in-place algorithm for solving a system of shift-XOR equations, called shift-XOR elimination, which does not have the bandwidth overhead generated by shift operations as in the previous zigzag algorithm and has lower computation complexities compared with the zigzag algorithm. The decoding and repair of shift-XOR MBR/MSR codes are decomposed into a sequence of systems of shift-XOR equations, and hence can be solved by a sequence of calls to the shift-XOR elimination. As the decompositions of the decoding and repair depend only on the PM construction, but not the specific shift and XOR operations, our decoding and repair schemes can be extended to other MBR/MSR codes using the PM construction. Due to its fundamental role, the shift-XOR elimination is of independent interest.

中文翻译：

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Shift-XOR 再生码的解码和修复方案

提出了用于移位异或（移位异或）乘积矩阵（PM）再生代码的解码和修复方案，其在通信和计算成本方面均优于现有方案。特别是，对于移位异或最小带宽再生 (MBR) 码，我们的解码和修复方案具有最佳传输带宽，并且可以就地实施，无需额外存储空间用于中间异或结果。从技术上讲，我们的方案涉及一种用于求解移位异或方程组的就地算法，称为移位异或消除，它没有像以前的锯齿形算法那样由移位操作产生的带宽开销，并且与之前的锯齿形算法相比，计算复杂度更低之字形算法。移位异或 MBR/MSR 码的解码和修复被分解为一系列移位异或方程组，因此可以通过对移位异或消除的调用序列来求解。由于解码和修复的分解仅依赖于 PM 构造，而不依赖于特定的移位和异或运算，因此我们的解码和修复方案可以使用 PM 构造扩展到其他 MBR/MSR 代码。由于其基本作用，shift-XOR 消除具有独立的意义。