  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 \emph{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.
