Analysis of Errors and Erasures in Parity Sharing RS Codecs

Abstract

Reed Solomon (RS) codes are widely used to protect information from errors in transmission and storage systems. Most of the RS codes are based on GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">8</sup> ) Galois Fields and use a byte to encode a symbol providing codewords up to 255 symbols. Codewords with more than 255 symbols can be obtained by using GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> ) Galois fields with m > 8, but this choice increases the complexity of the encoding and decoding algorithms. This limitation can be superseded by introducing parity sharing (PS) RS codes that are characterized by a greater flexibility in terms of design parameters. Consequently, a designer can choose between different PS code implementations in order to meet requirements such as bit error rate (BER), hardware complexity, speed, and throughput. This paper analyzes the performance of PS codes in terms of BER with respect to the code parameters, taking into account either random error or erasure rates as two independent probabilities. This approach provides an evaluation that is independent of the communication channel characteristics and extends the results to memory systems in which permanent faults and transient faults can be modeled, respectively, as erasures and random errors. The paper also provides hardware implementations of the PS encoder and decoder and discusses their performances in terms of hardware complexity, speed, and throughput.