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Fast nested Key Equation Solvers for Generalized Integrated Interleaved Decoder

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I-REGULAR PAPERS 2021年第1期68卷 483-495页

作者： Xie, Zhenshan Zhang, Xinmiao Ohio State Univ Dept Elect & Comp Engn Columbus OH 43210 USA

Generalized integrated interleaved (GII) codes nest Reed-Solomon (RS) or BCH sub-codewords to generate codewords belonging to stronger RS or BCH codes. Their hyper-speed decoding and good error-correction capability make them one of the best candidates for next-generation terabit/s digital storage and communications. The key equation solver (KES) in the nested decoding stage causes clock frequency bottleneck and takes a large portion of the GII decoder area. Recent architectures reduce the critical path to two multipliers and rely on the application of the slow-down technique to further reduce it to one. The slow-down technique requires two sub-codewords to be interleaved in the nested KES. However, most of the time, the nested decoding only needs to be carried out on one sub- codeword and half of the clock cycles are wasted. This paper proposes two fast nested KES algorithms, both of which have one multiplier in the critical path without applying slow-down and accordingly reduce the latency of the nested KES to almost a half. The short critical path is achieved by algorithmic reformulations that enable the pre-computation of the scalars in parallel with polynomial updating. Our second design adopts scaled versions of the polynomials to enable product term sharing so that the number of multipliers in each pair of processing elements is reduced from 8 as in the first design to 4. Novel scaling and combined scalar computations are developed to keep the critical path one multiplier. For an example GII code over GF(28) that has 3 nested codewords, our designs achieve 49.9% reduction on the number of clock cycles needed in the nested KES compared to prior designs. Besides, our second design requires 22% less area than the first one under the same timing constraint.

关键词： Error-correcting codes generalized integrated interleaved codes nested decoding key equation solver Reed-Solomon codes

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An Improved Method for Performance Analysis of Generalized Integrated Interleaved Codes

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IEEE COMMUNICATIONS LETTERS 2021年第10期25卷 3166-3169页

作者： Cui, Hangxuan Song, Suwen Wang, Zhongfeng Nanjing Univ Sch Elect Sci & Engn Nanjing 210008 Peoples R China

Generalized integrated interleaved (GII) codes show great potential in communication and distributed storage systems for their good error-correction capability and locality. To design GII codes with optimal code parameters, an accurate theoretical performance analysis method is essential, which also benefits the study of decoding behavior. However, the current performance analysis method does not fit simulation results very well and can only provide frame-error-rate (FER) evaluation. This letter presents an improved theoretical performance analysis method, including both bit-error-rate (BER) and FER estimations, which fit simulation results very closely. More importantly, the proposed method could predict the failure probability of each decoding round, thus facilitating a better understanding of decoding behavior. To simplify the analysis, an approximation performance evaluation method is also introduced by relaxing the constrained conditions, which still exhibits similar accuracy.

关键词： decoding Performance analysis Simulation Iterative decoding Estimation Encoding Software Generalized integrated interleaved (GII) codes locally recoverable codes (LRC) nested decoding theoretical performance analysis

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Scaled nested Key Equation Solver for Generalized Integrated Interleaved Decoder

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II-EXPRESS BRIEFS 2020年第11期67卷 2457-2461页

作者： Xie, Zhenshan Zhang, Xinmiao Ohio State Univ Dept Elect & Comp Engn Columbus OH 43210 USA

Generalized integrated interleaved (GII) codes nest short Reed-Solomon or BCH sub-codewords to form codewords of higher error-correction capability. They enable hyper-speed decoding and efficient failure recovery in distributed storage. Although the nested decoding can continue from the key equation solver (KES) result of the sub-codeword decoding, its discrepancy coefficient computation leads to long critical path. A recent nested KES architecture reduced the critical path to the same as that of the KES for sub-codeword decoding. Nevertheless, the nested KES has large area. This brief proposes a novel scaled nested KES algorithm and architecture to reduce the area requirement. Scaled versions of the polynomials are utilized to enable the sharing of product terms and the nested KES algorithm is reformulated to efficiently incorporate the scaling without changing the decoding output. As a result, the number of expensive finite field multipliers is significantly reduced. For an example code over $GF(2<^>{8})$ that corrects up to 28 errors, from architectural analysis, the proposed architecture achieves 17x0025;area reduction with only one more gate in the critical path.

关键词： decoding Computer architecture Matrix converters Iterative decoding Adders TV Error-correcting codes generalized integrated interleaved codes nested decoding key equation solver Reed-Solomon codes

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Efficient Architectures for Generalized Integrated Interleaved Decoder 52

Efficient Architectures for Generalized Integrated Interleav...

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IEEE International Symposium on Circuits and Systems (ISCAS)

作者： Zhang, Xinmiao Xie, Zhenshan Ohio State Univ Dept Elect & Comp Engn Columbus OH 43210 USA

ISBN: (纸本)9781728133201

Generalized integrated interleaved (GII) codes allow localized decoding of short sub-codewords. They are essential to hyper-speed data storage, communications, and continued scaling of distributed storage. Sub-codewords, which are usually Reed-Solomon (RS) or BCH codewords, are nested to generate codewords of higher correction capabilities. If the decoding of individual sub-codewords fails, the higher-order syndromes from the nested codewords are utilized to correct more errors. GII decoder design faces many challenges. The major ones include: 1) high-speed nested decoding utilizing the higher-order syndromes;2) efficient updating of higher-order syndromes after sub-codewords are corrected;3) computation of nested syndromes and matrix inversion for converting them to higher-order sub-codeword syndromes;and 4) efficient architectures capable of addressing the variable correction capabilities of the nested codewords. This paper proposes novel algorithmic reformulations and architectural transformations to address each bottleneck. For an example, GII code that has the same rate and length as eight un-nested (255, 223) RS codes, the proposed GII decoder achieves more than seven orders of magnitude improvement in error-correcting performance with less than 30% area overhead compared to the RS decoder. With a critical path of seven XOR gates, the proposed decoder can easily achieve more than 40 GByte/s throughput.

关键词： Error-correcting codes generalized integrated interleaved codes key equation solver locally recoverable codes nested decoding Reed-Solomon codes scalable Chien search syndromes Vandermonde matrix inversion

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Efficient Architectures for Generalized Integrated Interleaved Decoder

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I-REGULAR PAPERS 2019年第10期66卷 4018-4031页

作者： Zhang, Xinmiao Xie, Zhenshan Ohio State Univ Dept Elect & Comp Engn Columbus OH 43210 USA

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