In this paper, we explore new connections between the cycles in the graph of low-density parity-check (LDPC) codes and the eigenvalues of the corresponding adjacency matrix. The resulting observations are used to derive fast, simple, recursive formulas for the number of cycles N_2k of length 2k, k<g, in a bi-regular graph of girth g. Moreover, we derive explicit formulas for N_2k, k≤ 7, in terms of the nonzero eigenvalues of the adjacency matrix. Throughout, we focus on the practically interesting class of bi-regular quasi-cyclic LDPC (QC-LDPC) codes, for which the eigenvalues can be obtained efficiently by applying techniques used for block-circulant matrices.

Belief-propagation (BP) decoding for quantum low-density parity-check (QLDPC) codes is appealing due to its low complexity, yet it often exhibits convergence issues due to quantum degeneracy and short cycles that exist in the Tanner graph. To overcome this challenge, this paper proposes a reinforcement-learning (RL) approach that learns (offline) how to decode QLDPC codes based on sequential decoding trajectories. The decoding is formulated as a Markov decision process with a local, syndrome-driven state representation of the underlying RL agent. To enable fast inference, critical for practical implementation, we incrementally update our RL-based QLDPC decoder using second-order neighborhoods that avoid global rescans. Simulation results on representative QLDPC codes demonstrate the superiority of the proposed RL-based QLDPC decoders in terms of performance and convergence speed when compared to flooding and random sequential schedules, while achieving performance competitive with state-of-the-art BP-based decoders at comparable complexity.

Quantum low-density parity-check (QLDPC) codes are a leading approach to quantum error correction, yet conventional belief propagation (BP) decoders often perform poorly, primarily due to non-convergence exacerbated by stabilizer constraints, which induce short cycles and degeneracy. We propose two scheduling variants, sequential check node scheduling (SCNS) and sequential variable node scheduling (SVNS), that improve BP's error-correction ability by processing check nodes (CNs) or variable nodes (VNs), respectively, in a fixed order, stabilizing message updates and reducing stalls. We also employ this technique to an improved BP-variant called BP guided decimation (BPGD), where symbols are progressively fixed during decoding iterations. Here, we demonstrate that the sequential BPGD (SBPGD) decoder can further improve the convergence properties and performance of the decoder. On standard QLDPC benchmarks under a Pauli-X noise model, our sequential schedules are shown to lower the block error rate relative to conventional BP, and SBPGD outperforms BPGD while using significantly fewer decimation rounds, translating to lower computational cost. These results demonstrate that changing the update schedule, without altering the code, can improve both the reliability and efficiency of BP-based decoding for QLDPC codes. For the [[1922,50,16]] C2 hypergraph-product code with independent X errors, SVNS-BP surpasses BP-OSD-0 in error correction at roughly the same complexity as standard BP.

We investigate lossy source coding based on a soft-decision belief propagation guided decimation (BPGD) encoder for low-density generator matrix (LDGM) codes, referred to as soft-hard BPGD. The performance of this encoder is highly sensitive to the choice of “softness” parameters, typically denoted by (β,μ), which are conventionally tuned via exhaustive empirical sweeps. To reduce this burden and to better align the algorithm with the evolving graphical structure during decimation, we introduce a dynamic scheduling framework in which (β,μ) are not fixed globally but change as decimation progresses. The schedule starts in a softer regime to encourage exploration and gradually hardens toward the end to promote convergence, similar to simulated annealing. We consider linear and exponential schedules, discuss their physical interpretation via an effective temperature viewpoint, and explain how they integrate with soft-hard BPGD without changing the order of magnitude of its complexity. Numerical experiments with irregular and semi-regular LDGM ensembles indicate improved rate-distortion performance and reduced non-convergence compared to constant-parameter baselines, while largely eliminating expensive grid searches for a single best pair (β,μ).

Belief-propagation (BP) decoding for quantum low-density parity-check (QLDPC) codes is attractive due to its low complexity, but its performance is often limited by short cycles, degeneracy, and convergence failures. Recently, reinforcement-learning-based sequential variable-node (VN) scheduling (RL-S) was shown to improve BP decoding by learning state-dependent update orders. However, the VN-by-VN nature of that approach offers limited within-iteration parallelism, since only one VN is updated at a time. In this paper, we propose a cluster-based extension of RL-S for QLDPC codes. The VNs are partitioned into fixed clusters, and at each scheduling step the RL agent selects one cluster to update, after which all VNs in that cluster are updated in parallel using the same pre-update incoming messages. To keep the tabular state space practical for large cluster sizes, we introduce a permutation-invariant cluster state based on a normalized histogram of local mismatch weights, followed by quantization. This representation makes the number of cluster states depend on the quantization resolution rather than the cluster size. We also develop the corresponding cluster-level Markov decision process, reward function, and Q-learning update. Numerical results on representative QLDPC codes show that the proposed clustered learned scheduling preserves most of the error-rate benefit of VN-level learned sequential scheduling while substantially reducing the number of scheduling decisions per BP iteration, thereby providing an attractive latency-parallelism tradeoff.

This paper presents a finite-length polynomial-domain formulation of lifted-product quantum low-density parity-check (QLDPC) codes. We formulate the code construction over the quotient ring F 2[D]/(D L + 1), where polynomial base matrices are lifted entrywise to binary circulant blocks. This representation gives a compact algebraic description of the lifted-product parity-check matrices and allows CSS orthogonality to be analyzed before binary expansion. We show that the standard circulant lifting map is compatible with polynomial conjugation, which implies that the resulting binary matrices satisfy the CSS commutation constraint. The construction is illustrated with a constraint length 7, rate 1/2 NASA convolutional code example, and numerical examples are provided from a 3 x 4 polynomial parity-check matrix. The finite-length performance of selected constructed codes is then evaluated over the depolarizing channel using various benchmark decoders. The resulting framework gives a structured method to construct finite-length lifted-product QLDPC codes from small polynomial base matrices.

In this paper, we begin by reviewing the design of spatially coupled low-density parity-check (SC-LDPC) codes with belief- propagation based sliding window decoding (SWD) from the point of view of protograph constructions. We then discuss the potential for an SC-LDPC code to achieve better performance than the iterative decoding threshold of its underlying LDPC block code (LDPC-BC), subject to a constraint on decoding latency. In particular, we examine the problem of decoder error propagation that can occur under tight latency constraints and performance requirements and can severely degrade the decoded error rate. The existing literature on how to deal with this problem is then reviewed and the advantages and disadvantages of some proposed solutions are summarized, with emphasis on the code doping technique. Finally, various design tradeoffs and open research problems related to the ultimate goal of designing SC-LDPC codes with SWD to operate near capacity with moderate values of decoding latency, memory, and computational complexity are discussed, focusing on both the waterfall and error floor regions of the decoded error rate performance curve.

Generalized low-density parity-check (GLDPC) codes, where single parity-check constraints on the code bits are replaced with generalized constraints (an arbitrary linear code), are a promising class of codes for low-latency communication. The block error rate performance of the GLDPC codes, combined with a complementary outer code, has been shown to outperform a variety of state-of-the-art code and decoder designs with suitable lengths and rates for the 5G ultra-reliable low-latency communication (URLLC) regime. A major drawback of these codes is that it is not known how to construct appropriate polynomial matrices to encode them efficiently. In this paper, we analyze practical constructions of quasi-cyclic GLDPC (QC-GLDPC) codes and show how to construct polynomial generator matrices in various forms using minors of the polynomial matrix. The approach can be applied to fully generalized matrices or partially generalized (with mixed constraint node types) to find better performance/rate trade-offs. The resulting encoding matrices are presented in useful forms that facilitate efficient implementation. The rich substructure displayed also provides us with new methods of determining low weight codewords, providing lower and upper bounds on the minimum distance and often giving those of weight equal to the minimum distance. Based on the minors of the polynomial parity-check matrix, we also give a formula for the rank of any parity-check matrix representing a QC-LDPC or QC-GLDPC code, and hence, the dimension of the code. Finally, we show that by applying double graph-liftings, the code parameters can be improved without affecting the ability to obtain a polynomial generator matrix.

Belief-propagation (BP) decoding is attractive for quantum low-density parity-check (QLDPC) codes because it uses local message passing on sparse Tanner graphs. However, conventional flooding BP often stalls due to stabilizer degeneracy and short cycles. Reinforcement-learning-based sequential variable-node scheduling (RL-S), which learns the update order offline, has shown that adaptive scheduling can improve BP convergence. In this paper, we extend this idea with a second-order local update decoder, RL-S2LU. The proposed decoder preserves BP locality and low complexity, while numerical results show significant error-correction gains over conventional BP and the considered BP-OSD-10 baseline.

Spatially Coupled Low-Density Parity-Check (SCLDPC) codes are characterized by very long codeword lengths. For this reason, they are usually decoded with sliding window algorithms, which allow piecewise processing and decoding of the codeword symbols. In order to mitigate error propagation, it is possible to adapt strategies, such as non-uniform window sizes and node doping. In this paper, we propose a novel adaptive decoding schedule, which can be integrated with the aforementioned strategies. Numerical results confirm that the proposed approach can successfully detect error propagation events with more accuracy than conventional log-likelihood ratio-based approaches. Simulation results show that the error rate performance of time-invariant SC-LDPC codes significantly improves when the proposed strategies are adopted.

Short block-length polar-like codes showcase exceptional error correction performance (ECP) using sequential decoding or successive cancellation list decoding with a large list size. However, achieving a high level of reliability with these methods involves high-latency decoding. To meet the growing demand for low-latency communication with acceptable complexity, belief propagation (BP) decoding emerges as an attractive option, although its ECP is known to fall short of those high-latency alternatives. In this letter, we propose an enhanced BP decoding approach for polar codes, leveraging reinforcement learning (RL) to optimize the message-passing schedule. Moreover, we investigate the design of the polar code rate profile and corresponding Tanner graph representation to enhance the benefits of RL. Numerical results demonstrate a performance gain of more than 1 dB for polar codes with a length of 128 and a rate of 0.5 compared to conventional BP decoding alone at high E-b/N-0 values, demonstrating the promise of the proposed approach.

Despite advances in deep probabilistic models, learning discrete latent representations remains challenging. This work introduces a novel method to improve inference in discrete Variational Autoencoders by reframing the inference problem through a generative perspective. We conceptualize the model as a communication system, and propose to leverage Error-Correcting Codes (ECCs) to introduce redundancy in latent representations, allowing the variational posterior to produce more accurate estimates and reduce the variational gap. We present a proof-of-concept using a Discrete Variational Autoencoder with binary latent variables and low-complexity repetition codes, extending it to a hierarchical structure for disentangling global and local data features. Our approach significantly improves generation quality, data reconstruction, and uncertainty calibration, outperforming the uncoded models even when trained with tighter bounds such as the Importance Weighted Autoencoder objective. We also outline the properties that ECCs should possess to be effectively utilized for improved discrete variational inference.

In this paper, we study a new class of high-rate spatially coupled LDPC (SC-LDPC) codes based on the convolutional self-orthogonal codes (CSOCs) first introduced by Massey. The SC-LDPC codes are constructed by treating the irregular graph corresponding to the parity-check matrix of a systematic rate R = (n - 1)/n CSOC as a convolutional protograph. The protograph can then be lifted using permutation matrices to generate a high-rate SC-LDPC code whose strength depends on the lifting factor. The SC-LDPC codes constructed in this fashion can be decoded using iterative belief propagation (BP) based sliding window decoding (SWD). A non-systematic version of a CSOC parity-check matrix is then proposed by making a slight modification to the systematic construction. The non-systematic parity-check matrix corresponds to a regular protograph whose degree profile depends on the rate and error-correcting capability of the underlying CSOC. Even though the parity-check matrix is in non-systematic form, we show how systematic encoding can still be performed. We also show that the non-systematic convolutional protograph has a guaranteed girth and free distance and that these properties carry over to the lifted versions. Finally, numerical results are included demonstrating that CSOC-based SC-LDPC codes (i) achieve excellent performance at very high rates, (ii) have performance at least as good as that of SC-LDPC codes constructed from convolutional protographs commonly found in the literature, and (iii) have iterative decoding thresholds comparable to those of existing SC-LDPC code designs.

In this paper, we investigate ways to mitigate the problem of decoder error propagation (DEP) in sliding window decoding (SWD) of protograph-based spatially coupled low- density parity-check (SC-LDPC) codes for large frame length or streaming applications. In particular, in order to avoid subdividing a long frame into a series of shorter frames by using termination to combat DEP, we consider altering the code design by introducing occasional doped symbols into the encoded sequence, where the doping is accomplished by fixing the values of all or some of the variable nodes (VNs) at certain positions in the protograph.An important practical consideration in many applications is the ability to use systematic encoding. This necessitates that no more than a fraction of the VNs at any given position can be doped, i.e., that fractional doping be employed. We begin by showing numerically that full doping of a single position in the protograph of a long frame improves performance relative to that of terminating the frame at half its length. We then show that fractional doping of consecutive VN positions at a given location in the protograph is comparable to full doping. We also show that spreading fractionally doped positions over multiple locations in the protograph results in additional gains, where the effective code rate of a doped frame is always at least as high as that of the shorter terminated frame.

In this paper, we construct new quantum convolutional codes (QCCs) from classical self-orthogonal convolutional (CSOC) codes. We develop a methodology that achieves the required code orthogonality and allows for code performance optimization, generalizing to the full algebraic structure. This approach leverages the significant body of work in classical CSOCs to achieve a specified free distance.

This paper provides a cryptanalysis of a recently-proposed cryptosystem that uses a cascade of McEliece cryptosystems, the first using Goppa codes and the second using low-density parity-check (LDPC) codes. This cryptosystem decreases the key size while presuming to be as secure against known attacks as the original McEliece cryptosystem with Goppa codes. We show that the security of this specific cryptosystem reduces to the security of the McEliece cryptosystem instantiated with an LDPC code, upon which a key-recovery attack can be applied. As demonstrated in this paper, this reduction in security is enabled by an efficient attack against the sum of two codewords from two different codes whose generator matrices are available to the attacker. Combining this attack with the key-recovery attack, a key assumed to have 172-bit security loses 125 bits of security.

Due to code degeneracy and the graph structure of quantum low-density parity-check (QLDPC) codes, the performance of conventional belief propagation (BP) decoding can be poor. Recently, belief propagation guided decimation (BPGD) has shown promise to combat these challenges. In this paper, we investigate various decimation approaches to improve the error correcting performance and convergence speed of BPGD. We first consider soft decimation, where the BP equations are modified via several tuneable parameters. This approach exhibits linear complexity relative to the length of the block code and is shown to outperform hard decimation approaches for careful selection of the algorithm parameters. We then combine the approaches in a "soft-hard" BPGD variant, where hard decisions are periodically made and those symbols are permanently fixed throughout the remainder of the decoding process. Simulation results show that further performance improvement can be observed in this case at the cost of increasing the algorithmic complexity.

Low-density parity-check (LDPC) codes form part of the IRIG-106 standard and have been successfully deployed for the Telemetry Group version of shaped-offset quadrature phase shift keying (SOQPSK-TG) modulation. Recently, LDPC code solutions have been proposed and optimized for continuous phase modulations (CPMs), including pulse code modulation/frequency modulation (PCM/FM) and the multi-h CPM developed by the Advanced Range TeleMetry program (ARTM CPM). These codes were shown to perform around one dB from the respective channel capacities of these modulations. In this paper, we consider the effect of random puncturing and shortening of these LDPC codes to further improve spectrum efficiency. We perform asymptotic analyses of the ARTM0 code ensembles and present numerical simulation results that affirm the robust decoding performance promised by LDPC codes designed for ARTM CPM.

We propose a variant of the belief propagation guided decimation (BPGD) algorithm for the lossy binary sym-metric source coding problem, called DeciPolicy, which enables different decimation policies to decide when to trigger decimation, which variables to decimate, and which value to assign to decimated bits. In particular, we introduce a method that uses information about the cycles existing in the graph of a low-density generator matrix (LDGM) code to select candidate nodes for decimation. The proposed family of policies can be combined to include cycle detection-based decimation, parallel decimation of several bits, and random or hard value assignment. We demonstrate the algorithms on different constructions of LDGM codes, including an optimized irregular degree distribution and semi-regular Ising models, and show that our decimation policies lower the distortion when compared to various classical soft and hard BPGD algorithms, closing the gap to the rate-distortion limit.

Low-density parity-check (LDPC) codes form part of the IRIG-106 standard and have been successfully deployed for the Telemetry Group version of shaped-offset quadrature phase shift keying (SOQPSK-TG) modulation. Recently, LDPC code solutions have been proposed and optimized for continuous phase modulations (CPMs), including the pulse code modulation/frequency modulation (PCM/FM) and the multi-h CPM developed by the Advanced Range TeleMetry program (ARTM CPM). These codes were shown to perform around one dB from the respective channel capacities of these modulations. In this paper, we consider the effect of random puncturing of these LDPC codes to further improve spectrum efficiency. We present numerical simulation results that affirm the robust decoding performance promised by LDPC codes designed for ARTM CPM.