Fast En/Decoding of Reed-Solomon Codes for Failure Recovery

Abstract

Reed-Solomon (RS) codes are used in many storage systems for failure recovery. In popular software implementations, RS codes are defined by using a parity check matrix that is either a Cauchy matrix padded with an identity or a Vandermonde matrix. The encoding complexity can be reduced by searching for a Cauchy matrix that has a smaller number of ‘1's in its bit matrices or exploiting Reed-Muller (RM) transform in the Vandermonde matrix multiplication. This article proposes two new approaches that improve upon the previous schemes. In our first approach, different constructions of finite fields are explored to further reduce the number of ‘1's in the bit matrices of the Cauchy matrix and a new searching method is developed to find the matrices with minimum number of ‘1's. Our second approach defines RS codes using a parity check matrix in the format of a Vandermonde matrix concatenated with an identity matrix so that the multiplication with the inverse erasure columns in the encoding is eliminated and the decoding can be carried out using simplified formulas. The Vandermonde matrix in such an unconventional RS code definition needs to be constructed using finite field elements in non-consecutive order. A modification is also developed in this article to enable the application of the RM transform in this case to reduce the matrix multiplication complexity. For 4-erasure-correcting RS codes over <inline-formula><tex-math notation="LaTeX">$GF(2^8)$</tex-math></inline-formula> , the two proposed approaches increase the encoding throughput by 40 and 15 percent on average over the prior works based on Cauchy matrix and Vandermonde matrix with RM transform, respectively, for a range of codeword length. Moreover, the decoding throughput is also significantly improved.