Overhead-free in-place recovery and repair schemes of XOR-based regenerating codes

Abstract

Recovery and repair schemes are proposed for shift-exclusive-or (shift-XOR) product-matrix (PM) regenerating codes, which outperform those of the existing PM codes in terms of both communication and computation costs.In particular, for the minimum bandwidth regenerating (MBR) codes, our recovery and repair schemes have the optimal transmission bandwidth, zero decoding auxiliary data space and lower time complexity; for the minimum storage regenerating (MSR) codes, our recovery and repair schemes have smaller transmission bandwidth, smaller decoding auxiliary data space and lower time complexity. Moreover, our schemes involve only XOR operations in recovery and repair and are practical for system implementation. Technically, a procedure is first proposed for solving a system of shift-XOR equations, which plays a similar fundamental role as Gaussian elimination for solving systems of linear equations, and is of independent interest. The recovery and repair of shift-XOR MBR/MSR codes are then decomposed into a sequence of systems of shift-XOR equations, and hence can be solved by a sequence of calls to the procedure for solving a system of shift-XOR equations. As the decomposition of the recovery and repair depends only on the PM construction, but not the specific shift and XOR operations, our recovery and repair schemes can be extended to other MBR/MSR codes using the PM construction.

Application of Refined Increasing Difference Property in XOR-based MDS Codes

“…The encoding scheme of the distributed storage system in this paper follows that of [1], [10], [11]. The message consists of blocks, namely,…”

“…A block is a sequence of bits and the message contains bits. The generator matrix is an matrix, where ( ) satisfies the refined increasing-difference property [11]: for any , and such that and , where the first equality holds only when . (Normally, if , the matrix is a Vandermonde matrix, where it is in [1]).…”

“…The most well know generator matrix that satisfies the increasing difference property is the Vanermonde matrix. Then this property was applied to XOR-based regenerating codes [11].…”

“…This paper considers the XOR-based storage codes proposed in [10] due to the low encoding complexity. We combine the refined increasing difference property proposed in [11] and overhead-free in-place recovery scheme proposed in [1] together and apply the refined increasing difference property in XOR-based storage codes. Our scheme includes all the characteristics in [1] and we can obtain a lower storage complexity.…”