Method for correcting both errors and erasures of RS codes using error‐only and erasure‐only decoding algorithms

Abstract

A method for correcting both errors and erasures of Reed-Solomon (RS) codes using error-only and erasure-only decoding algorithms is proposed. First, the method removes the effect of erasures from syndromes, then it uses error-only and erasure-only correcting processes to correct errors and erasures, respectively; thus, the overall decoding complexity can be reduced substantially. Furthermore, with hardware implementation, the most time-consuming operations in these two processes can be calculated concurrently to obtain a significant reduction of decoding latency.Introduction: Reed-Solomon (RS) codes form a class of powerful random-error and multiple burst-error correcting codes such that they have been widely used in data communications and storage systems. Since (n, k) RS codes have the largest possible minimum distance d min = n − k + 1, the codes are classified as the class of maximum-distance-separable codes [1,2]. It is well-known that a code, with the minimum distance d min , has the capability of correcting all combinations of v symbol errors and e symbol erasures on the condition of 2v + e ≤ d min -1 [1-4], therefore three types of decoding algorithms can be derived: (i) for error-only correcting, (ii) for errorand-erasure correcting, and (iii) for erasure-only correcting. Among them, the erasure-only decoding type algorithms are the simplest and fastest because all erasure locations have been known. On the other hand, the error-and-erasure correcting type algorithms may be the most complicated and time-consuming because their syndrome polynomial, error-/erasure-location polynomial and error-/erasure-value evaluator have higher degrees than the error-only correcting type algorithms [5].For correcting both errors and erasures of RS codes, several wellknown methods can be found from [1][2][3][4]. In this Letter, we divide the error-and-erasure decoding procedure into two processes: (i) erroronly correcting and (ii) erasure-only correcting processes. The main operations in the error-only correcting process are to remove the effect of erasures from syndromes, as well as to find error locations and error values. The complexity of the error-only correcting process is less than that of the error-and-erasure correcting process because the effect of erasures has been removed, particularly for e≫v. The erasure-only correcting process corrects erasures using the modified syndromes which are refined from the original syndromes by removing the effect of errors. Since a complicated error-and-erasure correcting procedure has been divided into two simpler correcting processes, the overall decoding complexity can be reduced substantially. In addition, the most time-consuming operations in the erasure-only and error-only correcting processes can be computed concurrently by hardware circuits, so that the decoding delay will be efficiently reduced.

Highly Reliable Product Code for Error Correction

“…Next, we provide a specific comparison between error and erasures-and-errors decoding to show the advantage of erasures-and-errors decoding. An RS code with t error correction capability is capable of correcting v errors and e erasures, where 2v+e ≤ 2t [27]. In other words, the maximum correctable burst length (MCBL) by error decoding of RS codes is t, while the MCBL by erasures-and-errors decoding of RS codes is 2t.…”