Parallel Welch–Berlekamp Algorithm

Abstract

This paper presents new variants of the Welch-Berlekamp algorithm that are favorable to hardware implementation. First, we derive the parallel Welch-Berlekamp (PWB) algorithm in a constructive manner based on the properties of solutions to the rational interpolation problem. The algorithm features the simultaneously performed discrepancy computation and polynomial update. Second, we explore the early-termination mechanism of the PWB algorithm for decoding of Reed-Solomon (RS) codes. By introducing the concept of incomplete error locator polynomial, we show that if $e \leq t$ (where e is the number of errors and t is the error correction capability), the PWB algorithm can be terminated at latest at the completion of the $(t+ e)$ -th iteration. This leads to the early-terminating PWB (EPWB) algorithm. Finally, we develop frequency-domain versions of the PWB and EPWB algorithms, namely, FPWB and FEPWB. The key point toward the two algorithms is to replace the update of polynomial coefficients with the update of polynomial evaluations. It is worth noting that the FEPWB algorithm applies only to shortened RS codes. Furthermore, an efficient systolic architecture for the FPWB algorithm is designed, which is easily adapted for the FEPWB algorithm.