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1. Nested Array-Based Spatially Coupled LDPC Codes

   https://doi.org/10.1109/TCOMM.2021.3061689  

   Habib, Salman; Mitchell, David G.; Kliewer, Jorg (January 2021, IEEE Transactions on Communications)

   null (Ed.)

   Linear nested codes, where two or more sub-codes are nested in a global code, have been proposed as candidates for reliable multi-terminal communication. In this paper, we consider nested array-based spatially coupled low-density parity-check (SC-LDPC) codes and propose a line-counting based optimization scheme for minimizing the number of dominant absorbing sets in order to improve its performance in the high signal-to-noise ratio regime. Since the parity-check matrices of different nested sub-codes partially overlap, the optimization of one nested sub-code imposes constraints on the optimization of the other sub-codes. To tackle these constraints, a multi-step optimization process is applied first to one of the nested codes, then sequential optimization of the remaining nested codes is carried out based on the constraints imposed by the previously optimized sub-codes. Results show that the order of optimization has a significant impact on the number of dominant absorbing sets in the Tanner graph of the code, resulting in a trade-off between the performance of a nested code structure and its optimization sequence: the code which is optimized without constraints has fewer harmful structures than the code which is optimized with constraints. We also show that for certain code parameters, dominant absorbing sets in the Tanner graphs of all nested codes are completely removed using our proposed optimization strategy. 

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2. Increasing the Lifetime of Flash Memories Using Multi-Dimensional Graph-Based Codes

   https://doi.org/10.1109/ITW44776.2019.8989395  

   Hareedy, Ahmed; Kuditipudi, Rohith; Calderbank, Robert (August 2019, IEEE Information Theory Workshop (ITW 2019))

   In order to meet the demands of data-hungry applications, data storage devices are required to be increasingly denser. Various sources of error appear with this increase in density. Multi-dimensional (MD) graph-based codes are capable of mitigating error sources like interference and channel non-uniformity in dense storage devices. Recently, a technique was proposed to enhance the performance of MD spatially-coupled codes that are based on circulants. The technique carefully relocates circulants to minimize the number of short cycles. However, cycles become more detrimental when they combine together to form more advanced objects, e.g., absorbing sets, including low-weight codewords. In this paper, we show how MD relocations can be exploited to minimize the number of detrimental objects in the graph of an MD code. Moreover, we demonstrate the savings in the number of relocation arrangements earned by focusing on objects rather than cycles. Our technique is applicable to a wide variety of one-dimensional (OD) codes. Simulation results reveal significant lifetime gains in practical Flash systems achieved by MD codes designed using our technique compared with OD codes having similar parameters. 

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3. Scaled Fast Nested Key Equation Solver for Generalized Integrated Interleaved BCH Decoders

   https://doi.org/10.1109/ICASSP39728.2021.9414846  

   Xie, Zhenshan; Zhang, Xinmiao (June 2021, IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP))

   The generalized integrated interleaved BCH (GII-BCH) codes are among the best error-correcting codes for next-generation terabit/s memories. The key equation solver (KES) in the nested decoding of GII codes limits the achievable clock frequency. Recently, by polynomial scalar pre-computation, the critical path of the nested KES for Reed-Solomon (RS)-based GII codes has been reduced to one multiplier. However, for GII-BCH codes, the nested KES has more complicated formulas in order to skip the odd iterations and hence prior techniques do not directly extend. This paper proposes novel reformulations of the nested BCH KES to enable scalar pre-computation. Additionally, polynomial scaling is incorporated to enable complexity reduction. As a result, the critical path of the nested BCH KES with odd iterations skipped is reduced to one multiplier. For an example GII-BCH code over GF(2^{12}), the proposed design reduces the average nested BCH KES latency to around a half with similar silicon area compared to the best prior design. 

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4. Subsystem CSS codes, a tighter stabilizer-to-CSS mapping, and Goursat's Lemma

   https://doi.org/10.22331/q-2024-07-10-1403  

   Liu, Michael Liaofan; Tantivasadakarn, Nathanan; Albert, Victor V (July 2024, Quantum)

   The CSS code construction is a powerful framework used to express features of a quantum code in terms of a pair of underlying classical codes. Its subsystem extension allows for similar expressions, but the general case has not been fully explored. Extending previous work of Aly, Klappenecker, and Sarvepalli \cite{AKS06}, we determine subsystem CSS code parameters, express codewords, and develop a Steane-type decoder using only data from the two underlying classical codes. Generalizing a result of Kovalev and Pryadko \cite{KP13}, we show that any subsystem stabilizer code can be doubled to yield a subsystem CSS code with twice the number of physical, logical, and gauge qudits and up to twice the code distance. This mapping preserves locality and is tighter than the Majorana-based mapping of Bravyi, Terhal, and Leemhuis \cite{BTL10}. Using Goursat's Lemma, we show that every subsystem stabilizer code can be constructed from two nested subsystem CSS codes satisfying certain constraints, and we characterize subsystem stabilizer codes based on the nested codes' properties. 

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5. Optimizing quasi-cyclic spatially coupled LDPC codes by eliminating harmful objects

   https://doi.org/10.1186/s13638-023-02273-0  

   Battaglioni, Massimo; Chiaraluce, Franco; Baldi, Marco; Pacenti, Michele; Mitchell, David G. M. (July 2023, EURASIP Journal on Wireless Communications and Networking)

   Abstract It is well known that some harmful objects in the Tanner graph of low-density parity-check (LDPC) codes have a negative impact on their error correction performance under iterative message-passing decoding. Depending on the channel and the decoding algorithm, these harmful objects are different in nature and can be stopping sets, trapping sets, absorbing sets, or pseudocodewords. Differently from LDPC block codes, the design of spatially coupled LDPC codes must take into account the semi-infinite nature of the code, while still reducing the number of harmful objects as much as possible. We propose a general procedure, based onedge spreading, enabling the design of good quasi-cyclic spatially coupled LDPC (QC-SC-LDPC) codes. These codes are derived from quasi-cyclic LDPC (QC-LDPC) block codes and contain a considerably reduced number of harmful objects with respect to the original QC-LDPC block codes. We use an efficient way of enumerating harmful objects in QC-SC-LDPCCs to obtain a fast algorithm that spans the search space of potential candidates to select those minimizing the multiplicity of the target harmful objects. We validate the effectiveness of our method via numerical simulations, showing that the newly designed codes achieve better error rate performance than codes presented in previous literature. 

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