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1. 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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2. Using Minors to Construct Generator Matrices for Quasi-Cyclic LDPC Codes

   https://doi.org/10.1109/ISIT50566.2022.9834862  

   Smarandache, Roxana; Gomez-Fonseca, Anthony; Mitchell, David G. (June 2022, 2022 IEEE International Symposium on Information Theory (ISIT))

   This paper gives a simple method to construct generator matrices with polynomial entries (and hence offers an alternative encoding method to the one commonly used) for all quasi-cyclic low-density parity-check (QC-LDPC) codes, even for those that are rank deficient. The approach is based on constructing a set of codewords with the desired total rank by using minors of the parity-check matrix. We exemplify the method on several well-known and standard codes. Moreover, we explore the connections between the minors of the parity-check matrix and the known upper bound on minimum distance and provide a method to compute the rank of any parity-check matrix representing a QC-LDPC code, and hence the dimension of the code, by using the minors of the corresponding polynomial parity-check matrix. 

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3. Concatenated Spatially Coupled LDPC Codes with Sliding Window Decoding for Joint Source-Channel Coding

   https://doi.org/10.1109/tcomm.2021.3126750  

   Golmohammadi, Ahmad; Mitchell, David G. (November 2021, IEEE Transactions on Communications)

   In this paper, a method for joint source-channel coding (JSCC) based on concatenated spatially coupled low-density parity-check (SC-LDPC) codes is investigated. A construction consisting of two SC-LDPC codes is proposed: one for source coding and the other for channel coding, with a joint belief propagation-based decoder. Also, a novel windowed decoding (WD) scheme is presented with significantly reduced latency and complexity requirements. The asymptotic behavior for various graph node degrees is analyzed using a protograph-based Extrinsic Information Transfer (EXIT) chart analysis for both LDPC block codes with block decoding and for SC-LDPC codes with the WD scheme, showing robust performance for concatenated SC-LDPC codes. Simulation results show a notable performance improvement compared to existing state-of-the-art JSCC schemes based on LDPC codes with comparable latency and complexity constraints. 

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4. Information Bottleneck Decoding of Rate-Compatible 5G-LDPC Codes

   https://doi.org/10.1109/ICC40277.2020.9149304  

   Stark, Maximilian; Bauch, Gerhard; Wang, Linfang; Wesel, Richard D. (June 2020, ICC 2020 - 2020 IEEE International Conference on Communications (ICC))

   The new 5G communications standard increases data rates and supports low-latency communication that places constraints on the computational complexity of channel decoders. 5G low-density parity-check (LDPC) codes have the so-called protograph-based raptor-like (PBRL) structure which offers inherent rate-compatibility and excellent performance. Practical LDPC decoder implementations use message-passing decoding with finite precision, which becomes coarse as complexity is more severely constrained. Performance degrades as the precision becomes more coarse. Recently, the information bottleneck (IB) method was used to design mutual-information-maximizing lookup tables that replace conventional finite-precision node computations. The IB approach exchanges messages represented by integers with very small bit width. This paper extends the IB principle to the flexible class of PBRL LDPC codes as standardized in 5G. The extensions include puncturing and rate-compatible IB decoder design. As an example of the new approach, a 4-bit information bottleneck decoder is evaluated for PBRL LDPC codes over a typical range of rates. Frame error rate simulations show that the proposed scheme outperforms offset min-sum decoding algorithms and operates very close to double-precision sum-product belief propagation decoding. 

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5. Matrix Multiplication Verification Using Coding Theory

   https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.42  

   Bennett, Huck; Gajulapalli, Karthik; Golovnev, Alexander; Warton, Evelyn (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Kumar, Amit; Ron-Zewi, Noga (Ed.)

   We study the Matrix Multiplication Verification Problem (MMV) where the goal is, given three n × n matrices A, B, and C as input, to decide whether AB = C. A classic randomized algorithm by Freivalds (MFCS, 1979) solves MMV in Õ(n²) time, and a longstanding challenge is to (partially) derandomize it while still running in faster than matrix multiplication time (i.e., in o(n^ω) time). To that end, we give two algorithms for MMV in the case where AB - C is sparse. Specifically, when AB - C has at most O(n^δ) non-zero entries for a constant 0 ≤ δ < 2, we give (1) a deterministic O(n^(ω-ε))-time algorithm for constant ε = ε(δ) > 0, and (2) a randomized Õ(n²)-time algorithm using δ/2 ⋅ log₂ n + O(1) random bits. The former algorithm is faster than the deterministic algorithm of Künnemann (ESA, 2018) when δ ≥ 1.056, and the latter algorithm uses fewer random bits than the algorithm of Kimbrel and Sinha (IPL, 1993), which runs in the same time and uses log₂ n + O(1) random bits (in turn fewer than Freivalds’s algorithm). Our algorithms are simple and use techniques from coding theory. Let H be a parity-check matrix of a Maximum Distance Separable (MDS) code, and let G = (I | G') be a generator matrix of a (possibly different) MDS code in systematic form. Our deterministic algorithm uses fast rectangular matrix multiplication to check whether HAB = HC and H(AB)^T = H(C^T), and our randomized algorithm samples a uniformly random row g' from G' and checks whether g'AB = g'C and g'(AB)^T = g'C^T. We additionally study the complexity of MMV. We first show that all algorithms in a natural class of deterministic linear algebraic algorithms for MMV (including ours) require Ω(n^ω) time. We also show a barrier to proving a super-quadratic running time lower bound for matrix multiplication (and hence MMV) under the Strong Exponential Time Hypothesis (SETH). Finally, we study relationships between natural variants and special cases of MMV (with respect to deterministic Õ(n²)-time reductions). 

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