In this paper we devise a rational curve fitting algorithm and apply it to the list decoding of Reed-Solomon and BCH codes. The resulting list decoding algorithms exhibit the following significant properties.• The algorithm achieves the limit of list error correction capability (LECC) n(1 − √ 1 − D)for a (generalized) (n, k, d = n − k + 1) Reed-Solomon code, which matches the Johnson bound, where D △ = d n denotes the normalized minimum distance. The algorithmic complexity is O n 6 (1 − √ 1 − D) 8 . In comparison with the Guruswami-Sudan algorithm, which exhibits the same LECC, the proposed requires a multiplicity (which dictates the algorithmic complexity) significantly smaller than that of the Guruswami-Sudan algorithm in achieving a given LECC, except for codes with code-rate below 0.15. In particular, for medium-to-high rate codes, the proposed algorithm reduces the multiplicity by orders of magnitude. Moreover, for any ǫ > 0, the intermediate LECC t = ⌊ǫ · d 2 + (1 − ǫ) · (n − n(n − d))⌋ can be achieved by the proposed algorithm with multiplicity m = ⌊ 1 ǫ ⌋. Its list size is shown to be upper bounded by a constant with respect to a fixed normalized minimum distance D, rendering the algorithmic complexity quadratic in nature, O(n 2 ).• By utilizing the unique properties of the Berlekamp algorithm, the algorithm achieves the LECC limit n 2 (1 − √ 1 − 2D) for a narrow-sense (n, k, d) binary BCH code, which matches the Johnson bound for binary codes. The algorithmic complexity is O n 6 (1 − √ 1 − 2D) 8 . Moreover, for any ǫ > 0, the intermediate LECC t = ⌊ǫ · d 2 + (1 − ǫ) · n− √ n(n−2d) 2 ⌋ can be achieved by the proposed algorithm with multiplicity m = ⌊ 1 2ǫ ⌋. Its list size is shown to be upper bounded by a constant, rendering the algorithmic complexity quadratic in nature, O(n 2 ).
We derive the Wu list-decoding algorithm for Generalised Reed-Solomon (GRS) codes by using Gröbner bases over modules and the Euclidean algorithm (EA) as the initial algorithm instead of the Berlekamp-Massey algorithm (BMA). We present a novel method for constructing the interpolation polynomial fast. We give a new application of the Wu list decoder by decoding irreducible binary Goppa codes up to the binary Johnson radius. Finally, we point out a connection between the governing equations of the Wu algorithm and the Guruswami-Sudan algorithm (GSA), immediately leading to equality in the decoding range and a duality in the choice of parameters needed for decoding, both in the case of GRS codes and in the case of Goppa codes.
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.
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