New Scalable Decoder Architectures for Reed–Solomon Codes

Abstract

In this paper, we devise new scalable decoder architectures for Reed-Solomon (RS) codes, comprising three parts: error-only decoding, error-erasure decoding, and their decoding for singly extended RS codes. New error-only decoders are devised through algorithmic transformations of the inversionless Berlekamp-Massey algorithm (IBMA). We first generalize the Horiguchi-Koetter formula to evaluate error magnitudes using the error locator polynomial (x) and the auxiliary polynomial B(x) produced by IBMA, which effectively eliminates the computation of error evaluator polynomial. We next devise an enhanced parallel inversionless Berlekamp-Massey algorithm (ePIBMA) that effectively takes advantage of the generalized Horiguchi-Koetter formula. The derivative ePIBMA architecture requires only 2t + 1 (t denotes the error correction capability) systolic cells, in contrast with 3t or more cells of the existing regular architectures based on IBMA or the Euclidean algorithm. Moreover, it may literally function as a linear-feedback-shift-register encoder. New error-erasure decoders are devised through algorithmic transformations of the inversionless Blahut algorithm (IBA). The proposed split parallel inversionless Blahut algorithm (SPIBA) yields merely 2t + 1 systolic cells, which is the same number as the error-only decoder ePIBMA. The task is partitioned into two separate steps, computing the complementary errorerasure evaluator polynomial followed by computing error-erasure locator polynomial, both utilizing SPIBA. Surprisingly, it has exactly the same number of cells and literally the same complexity and throughput as the proposed error-only decoder architecture ePIBMA; it employs 33% less hardware and at the same time achieves more than twice faster throughput, than the serial architecture IBA. we further propose a unified parallel inversionless Blahut algorithm (UPIBA) by incorporating the key virtues of the error-only decoder ePIBMA into SPIBA. The complexity and throughput of the rderivative UPIBA architecture are literally the same as ePIBMA and SPIBA, while performing almost equally efficiently as ePIBMA on error-only decoding and as SPIBA on error-erasure decoding. UPIBA also inherits the dynamic power saving feature of ePIBMA and SPIBA. Indeed, UPIBA renders highly attractive for on-the-fly implementation of error-erasure decoding. We finally demonstrate that the proposed decoders, i.e., ePIBMA, SPIBA, and UPIBA, can be magically migrated to decode singly extended RS codes, with negligible add-ons, except that an extra multiplexer is added to their critical paths. To the author's best knowledge, this is the first time that a high-throughput decoder for singly extended RS codes is explored.

VLSI Architectures for Reed–Solomon Codes: Classic, Nested, Coupled, and Beyond

“…The complexity of the KES is further reduced from 3t + 1 PEs as in the riBM and RiBM architectures to 2t + 1 PEs in the enhanced Parallel Inversionless (ePI-) BM architecture [27]. The reduction is enabled by using the Horiguchi-Koetter formula to compute the error magnitudes as…”

“…The ePIBM algorithm is listed in Algorithm C and the corresponding implementation architecture is shown in Fig. 6 [27]. To better associate the architecture with the polynomial updating, Λ(x) and B(x) are integrated into the∆(x) and Θ(x) polynomials, respectively, in Algorithm C. The x 2t terms in the∆ (0) 2t−1 = 0 is also reflected in theΘ (0) (x) initialization.…”

“…To better associate the architecture with the polynomial updating, Λ(x) and B(x) are integrated into the∆(x) and Θ(x) polynomials, respectively, in Algorithm C. The x 2t terms in the∆ (0) 2t−1 = 0 is also reflected in theΘ (0) (x) initialization. By making this modification, it has been proved in [27]…”

“…If B(x) is used to compute the error magnitudes, the complexity of the following Chien search becomes much higher. To address this issue, it was found in [27] that if the number of errors does not exceed t and L (K) B reaches t − 1 in an iteration r = K, then the inverse roots of Λ (K+1) (x) are already the error locators. As a result, the error magnitudes can be computed as…”

Soft-Decision Low-Complexity Chase Decoders for the RS(255,239) Code

“…The third one, which we refer to as Q6, is a quasi-η = 6 LCC: it uses all the test vectors of a true η = 6 LCC, but four. They are based on a systolic KES, the enhanced parallel inversionless Berlekamp-Massey algorithm (ePiBMA) [22], that requires 2t = 16 cycles for the computation of each frame, with low critical path: one adder (T + ), one multiplexer (T x ) and one multiplier (T * ). Moreover, the selected KES requires fewer resources than other popular options.…”