# Decoding and Repair Schemes for Shift-XOR Regenerating Codes

## TL;DR

This paper introduces efficient decoding and repair schemes for shift-XOR regenerating codes that reduce communication and computation costs, featuring an in-place algorithm called shift-XOR elimination with broad applicability.

## Contribution

The paper presents a novel in-place shift-XOR elimination algorithm for decoding and repairing shift-XOR PM regenerating codes, improving efficiency and extending to other code types.

## Key findings

- Optimal transmission bandwidth for shift-XOR MBR codes.
- Lower computational complexity compared to previous zigzag algorithms.
- Schemes can be extended to other PM-based MBR/MSR codes.

## Abstract

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.