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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. 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)

   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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5. Optimizing Finite-Blocklength Nested Linear Secrecy Codes: Using the Worst Code to Find the Best Code

   https://doi.org/10.3390/e25101456  

   Shoushtari, Morteza ; Harrison, Willie ( October 2023 , Entropy)

   Nested linear coding is a widely used technique in wireless communication systems for improving both security and reliability. Some parameters, such as the relative generalized Hamming weight and the relative dimension/length profile, can be used to characterize the performance of nested linear codes. In addition, the rank properties of generator and parity-check matrices can also precisely characterize their security performance. Despite this, finding optimal nested linear secrecy codes remains a challenge in the finite-blocklength regime, often requiring brute-force search methods. This paper investigates the properties of nested linear codes, introduces a new representation of the relative generalized Hamming weight, and proposes a novel method for finding the best nested linear secrecy code for the binary erasure wiretap channel by working from the worst nested linear secrecy code in the dual space. We demonstrate that our algorithm significantly outperforms the brute-force technique in terms of speed and efficiency.

    

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