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1. Optimization of Nested Array-based LDPC Codes Via Spatial Coupling

   https://doi.org/10.1109/ITW44776.2019.8989313  

   Habib, Salman ; Mitchell, David G. ; Kliewer, Jorg ( August 2019 , 2019 IEEE Information Theory Workshop (ITW))

   Linear nested codes, where two or more subcodes 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 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. The presented multi-step optimization process is applied first to one of the nested codes, then an optimization of the remaining nested codes is carried out based on these code constraints. We also show that for certain code parameters, dominant absorbing sets in the Tanner graphs of all nested codes can be completely removed using our proposed optimization strategy. 

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2. 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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3. Single-Shot Decoding of Good Quantum LDPC Codes

   https://doi.org/10.1007/s00220-024-04951-6  

   Gu, Shouzhen ; Tang, Eugene ; Caha, Libor ; Choe, Shin Ho ; He, Zhiyang ; Kubica, Aleksander ( March 2024 , Communications in Mathematical Physics)

   Quantum Tanner codes constitute a family of quantum low-density parity-check codes with good parameters, i.e., constant encoding rate and relative distance. In this article, we prove that quantum Tanner codes also facilitate single-shot quantum error correction (QEC) of adversarial noise, where one measurement round (consisting of constant-weight parity checks) suffices to perform reliable QEC even in the presence of measurement errors. We establish this result for both the sequential and parallel decoding algorithms introduced by Leverrier and Zémor. Furthermore, we show that in order to suppress errors over multiple repeated rounds of QEC, it suffices to run the parallel decoding algorithm for constant time in each round. Combined with good code parameters, the resulting constant-time overhead of QEC and robustness to (possibly time-correlated) adversarial noise make quantum Tanner codes alluring from the perspective of quantum fault-tolerant protocols.

    

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4. On the Design of Generalized LDPC Codes with Component BCJR Decoding

   https://doi.org/10.1109/GLOBECOM42002.2020.9322143  

   Liu, Yanfang ; Olmos, Pablo M. ; Mitchell, David G. ( December 2020 , GLOBECOM 2020 - 2020 IEEE Global Communications Conference)

   null (Ed.)

   Generalized low-density parity-check (GLDPC) codes, where the single parity-check (SPC) nodes are replaced by generalized constraint (GC) nodes, are known to offer a reduced gap to capacity when compared with conventional LDPC codes, while also maintaining linear growth of minimum distance. However, for certain classes of practical GLDPC codes, there remains a gap to capacity even when utilizing blockwise decoding algorithm at GC nodes. In this work, we propose to optimize the design of GLDPC codes where the GC nodes are decoded with a trellis-based bit-wise Bahl-Cocke-Jelinek- Raviv (BCJR) component decoding algorithm. We analyze the asymptotic threshold behavior of GLDPC codes and determine the optimal proportion of the GC nodes in the GLDPC Tanner graph.We show significant performance improvements compared to existing designs with the same order of decoding complexity. 

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5. On the Tanner Cycle Distribution of QC-LDPC Codes from Polynomial Parity-Check Matrices

   https://doi.org/10.1109/isit54713.2023.10207005  

   Gómez-Fonseca, Anthony ; Smarandache, Roxana ; Mitchell, David G. ( June 2023 , 023 IEEE International Symposium on Information Theory (ISIT))

   In this paper, we present an efficient strategy to enumerate the number of k-cycles, g ≤ k < +2g, in the Tanner graph of a quasi-cyclic low-density parity-check (QC-LDPC) code with girth g using its polynomial parity-check matrix H. This strategy works for both (n c , n v )-regular and irregular QC-LDPC codes. In this approach, we note that the mth power of the polynomial adjacency matrix can be used to describe walks of length m in the protograph and can therefore be sufficiently described by the matrices Bm(H)≜(HH⊤)⌊m/2⌋H(mmod2), where m ≥ 0. For example, in the case of QC-LDPC codes based on the 3 × n v fully-connected protograph, the complexity of determining the number of k-cycles, Nk, for k = 4, 6 and 8, is O(n2vlog(N)), O(n2vlog(nv)log(N)) and O(n4vlog4(nv)log(N)), respectively. The complexity, depending logarithmically on the lifting factor N, gives our approach, to the best of our knowledge, a significant advantage over previous works on the cycle distribution of QC-LDPC codes. 

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