arXiv subjects

David G. M. Mitchell

Publications and source records attributed to David G. M. Mitchell.

At least 19 recordsLinked to original sources

arXiv2026

Lifted-Product QLDPC Codes in the Polynomial Domain

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.

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arXiv2026

High-Performance Reinforcement-Learned BP Decoding of Quantum LDPC Codes

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.

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arXiv2026

Learning to Decode Quantum LDPC Codes via Cluster-Based Sequential Belief Propagation

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.

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arXiv2026

Dynamic Parameter Scheduling in Soft-Hard BPGD for Lossy Source Coding

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 \emph{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 \emph{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 $(β,μ)$.

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arXiv2026

Learning to Decode Quantum LDPC Codes Via Belief Propagation

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.

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arXiv2026

Sequential BP-based Decoding of QLDPC Codes

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.

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arXiv2026

The Number of Cycles of Bi-regular Tanner Graphs in Terms of the Eigenvalues of the Adjacency Matrix

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 $N_{2k}$, $k\leq 7$, 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.

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arXiv2025

Generalized Quasi-Cyclic LDPC Codes: Design and Efficient Encoding

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.

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arXiv2025

High-Rate Spatially Coupled LDPC Codes Based on Massey's Convolutional Self-Orthogonal Codes

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.

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arXiv2024

Spectrally Efficient LDPC Codes For IRIG-106 Waveforms via Random Puncturing

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.

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arXiv2024

PAC Code Rate-Profile Design Using Search-Constrained Optimization Algorithms

In this paper, we introduce a novel rate-profile design based on search-constrained optimization techniques to assess the performance of polarization-adjusted convolutional (PAC) codes under Fano (sequential) decoding. The results demonstrate that the resulting PAC code offers much reduced computational complexity compared to a construction based on a conventional genetic algorithm without a performance loss in error-correction performance. As the fitness function of our algorithm, we propose an adaptive successive cancellation list decoding algorithm to determine the weight distribution of the rate profiles. The simulation results indicate that, for a PAC(256, 128) code, only 8% of the population requires that their fitness function be evaluated with a large list size. This represents an improvement of almost 92% over a conventional evolutionary algorithm. For a PAC(64, 32) code, this improvement is about 99%. We also plotted the performance of the high-rate PAC(128, 105) and PAC(64, 51) codes, and the results show that they exhibit superior performance compared to other algorithms.

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arXiv2023

Reinforcement Learning for Sequential Decoding of Generalized LDPC Codes

In this work, we propose reinforcement learning (RL) for sequential decoding of moderate length generalized low-density parity-check (GLDPC) codes. Here, sequential decoding refers to scheduling all the generalized constraint nodes (GCNs) and single parity-check nodes (SPCNs) of a GLDPC code serially in each iteration. A GLDPC decoding environment is modeled as a finite Markov decision process (MDP) in which the state-space comprises of all possible sequences of hard-decision values of the variables nodes (VNs) connected to the scheduled GCN or SPCN, and the action-space of the MDP consists of all possible actions (GCN and SPCN scheduling). The goal of RL is to determine an optimized scheduling policy, i.e., one that results in a decoded codeword by minimizing the complexity of the belief propagation (BP) decoder. For training, we consider the proportion of correct bits at the output of the GCN or SPCN as a reward once it is scheduled. The expected rewards for scheduling all the GCNs/SPCNs in the code's Tanner graph are earned via BP decoding during the RL phase. The proposed RL-based decoding scheme is shown to significantly outperform the standard BP flooding decoder, as well as a sequential decoder in which the GCNs/SPCNs are scheduled randomly.

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arXiv2021

Iterative Threshold Decoding of Spatially Coupled, Parallel-Concatenated Codes

Spatially coupled, parallel concatenated codes (SC-PCCs) have been shown to approach channel capacity when decoded using optimal iterative methods. However, under complexity constraints such decoding strategies can result in unacceptable power and latency costs. In this work, we employ convolutional self-orthogonal component codes along with low-complexity, suboptimal a posteriori probability (APP) threshold decoders with SC-PCCs to reduce decoding complexity. The proposed code design is faster, more energy efficient, and easier to implement than optimal methods, while offering significant coding gain over existing threshold decodable, turbo-like constructions of similar latency and complexity. The design also serves to further illustrate the advantages spatial coupling can provide to existing code constructions and decoder implementations.

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arXiv2021

A Unifying Framework to Construct QC-LDPC Tanner Graphs of Desired Girth

This paper presents a unifying framework to construct low-density parity-check (LDPC) codes with associated Tanner graphs of desired girth. Towards this goal, we highlight the role that a certain square matrix that appears in the product of the parity-check matrix with its transpose has in the construction of codes with graphs of desired girth and further explore it in order to generate the set of necessary and sufficient conditions for a Tanner graph to have a given girth between 6 and 12. For each such girth, we present algorithms to construct codes of the desired girth and we show how to use them to compute the minimum necessary value of the lifting factor. For girth larger than 12, we show how to use multi-step graph lifting methods to deterministically modify codes in order to increase their girth. We also give a new perspective on LDPC protograph-based parity-check matrices by viewing them as rows of a parity-check matrix equal to the sum of certain permutation matrices and obtain an important connection between all protographs and those with variable nodes of degree 2. We also show that the results and methodology that we develop for the all-one protograph can be used and adapted to analyze the girth of the Tanner graph of any parity-check matrix and demonstrate how this can be done using a well-known irregular, multi-edge protograph specified by the NASA Consultative Committee for Space Data Systems (CCSDS). Throughout the paper, we exemplify our theoretical results with constructions of LDPC codes with Tanner graphs of any girth between 6 and 14 and give sufficient conditions for a multi-step lifted parity-check matrix to have girth between 14 and 22.

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arXiv2021

Necessary and Sufficient Girth Conditions for LDPC Tanner Graphs with Denser Protographs

This paper gives necessary and sufficient conditions for the Tanner graph of a quasi-cyclic (QC) low-density parity-check (LDPC) code based on the all-one protograph to have girth 6, 8, 10, and 12, respectively, in the case of parity-check matrices with column weight 4. These results are a natural extension of the girth results of the already-studied cases of column weight 2 and 3, and it is based on the connection between the girth of a Tanner graph given by a parity-check matrix and the properties of powers of the product between the matrix and its transpose. The girth conditions can be easily incorporated into fast algorithms that construct codes of desired girth between 6 and 12; our own algorithms are presented for each girth, together with constructions obtained from them and corresponding computer simulations. More importantly, this paper emphasizes how the girth conditions of the Tanner graph corresponding to a parity-check matrix composed of circulants relate to the matrix obtained by adding (over the integers) the circulant columns of the parity-check matrix. In particular, we show that imposing girth conditions on a parity-check matrix is equivalent to imposing conditions on a square circulant submatrix of size 4 obtained from it.

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arXiv2021

Necessary and Sufficient Girth Conditions for Tanner Graphs of Quasi-Cyclic LDPC Codes

This paper revisits the connection between the girth of a protograph-based LDPC code given by a parity-check matrix and the properties of powers of the product between the matrix and its transpose in order to obtain the necessary and sufficient conditions for a code to have given girth between 6 and 12, and to show how these conditions can be incorporated into simple algorithms to construct codes of that girth. To this end, we highlight the role that certain submatrices that appear in these products have in the construction of codes of desired girth. In particular, we show that imposing girth conditions on a parity-check matrix is equivalent to imposing conditions on a square submatrix obtained from it and we show how this equivalence is particularly strong for a protograph based parity-check matrix of variable node degree 2, where the cycles in its Tanner graph correspond one-to-one to the cycles in the Tanner graph of a square submatrix obtained by adding the permutation matrices (or products of these) in the composition of the parity-check matrix. We end the paper with exemplary constructions of codes with various girths and computer simulations. Although, we mostly assume the case of fully connected protographs of variable node degree 2 and 3, the results can be used for any parity-check matrix/protograph-based Tanner graph.

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arXiv2021

Spatially Coupled Generalized LDPC Codes: Asymptotic Analysis and Finite Length Scaling

Generalized low-density parity-check (GLDPC) codes are a class of LDPC codes in which the standard single parity check (SPC) constraints are replaced by constraints defined by a linear block code. These stronger constraints typically result in improved error floor performance, due to better minimum distance and trapping set properties, at a cost of some increased decoding complexity. In this paper, we study spatially coupled generalized low-density parity-check (SC-GLDPC) codes and present a comprehensive analysis of these codes, including: (1) an iterative decoding threshold analysis of SC-GLDPC code ensembles demonstrating capacity approaching thresholds via the threshold saturation effect; (2) an asymptotic analysis of the minimum distance and free distance properties of SC-GLDPC code ensembles, demonstrating that the ensembles are asymptotically good; and (3) an analysis of the finite-length scaling behavior of both GLDPC block codes and SC-GLDPC codes based on a peeling decoder (PD) operating on a binary erasure channel (BEC). Results are compared to GLDPC block codes, and the advantages and disadvantages of SC-GLDPC codes are discussed.

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arXiv2021

Nested Array-Based Spatially Coupled LDPC Codes

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 tradeoff 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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