A tutorial on Reed–Solomon coding for fault‐tolerance in RAID-like systems

Abstract

It is well-known that Reed-Solomon codes may be used to provide error correction for multiple failures in RAIDlike systems. The coding technique itself, however, is not as well-known. To the coding theorist, this technique is a straightforward extension to a basic coding paradigm and needs no special mention. However, to the systems programmer with no training in coding theory, the technique may be a mystery. Currently, there are no references that describe how to perform this coding that do not assume that the reader is already well-versed in algebra and coding theory. This paper is intended for the systems programmer. It presents a complete specification of the coding algorithm plus details on how it may be implemented. This specification assumes no prior knowledge of algebra or coding theory. The goal of this paper is for a systems programmer to be able to implement Reed-Solomon coding for reliability in RAID-like systems without needing to consult any external references. Problem Specification Let there be storage devices, ¡ £ ¢ ¥ ¤ ¦ ¡ § © ¤ ¤ ¦ ¡ , each of which holds bytes. These are called the "Data Devices." Let there be more storage devices ¢ ¥ ¤ § ¤ ¤ , each of which also holds bytes. These are called the "Checksum Devices." The contents of each checksum device will be calculated from the contents of the data devices. The goal is to define the calculation of each " ! such that if any of ¡ ¢¤ ¦ ¡ §¤ ¤ ¦ ¡ ¤ ¢¤ §¤ ¤ fail, then the contents of the failed devices can be reconstructed from the non-failed devices.

Gibraltar: A Reed‐Solomon coding library for storage applications on programmable graphics processors

“…The primary operation in Reed-Solomon coding is the multiplication of F , the lower m rows of an information dispersal matrix A D Ä I F , with a vector of data elements d [27].…”

“…x y D exp.log.x/ C log.y// (5) where the addition operator denotes normal integer addition modulo 2 w 1, whereas exp./ and log./ are, respectively, exponentiation and logarithm operations in the finite field using a common base [27]. Because w D 8 for RAID systems, an implementation can contain pre-calculated tables for the exp and log operations, which are each 256 B in length.…”

“…Unfortunately, parallel table lookup capability is not common, and CPU implementations of finite field arithmetic tend to suffer accordingly. A more in-depth treatment of implementation details of Reed-Solomon coding is available [27].…”

“…The properties of finite field arithmetic and the methods of generation for A will ensure that these equalities hold, A 0 is invertible, and all elements generated are within the bounds specified by the domain of the code (e.g., 8-bit values) [27].…”

A Comparison of Several Fault-Tolerance Methods for the Detection and Correction of Floating-Point Errors in Matrix-Matrix Multiplication

“…Both ABFT and RC are known to enable error detection, but ABFT has received much more attention because it is also deployed for error correction. In theory, ABFT can correct up to k errors with 2k + 1 checksums [16,19,20]. However, the numerical instability of floating-point ABFT currently limits its usage to correct one or two errors within a kernel.…”