Quantum low-density parity-check (QLDPC) codes with asymptotically non-zero rates are prominent candidates for achieving fault-tolerant quantum computation, primarily due to the low operational depth of their syndrome-measurement circuits. Numerous studies advocate the necessity of fast decoders to fully harness the capabilities of QLDPC codes, thus driving the focus towards designing low-complexity iterative decoders. However, empirical investigations indicate that such iterative decoders are susceptible to having a high error floor when decoding QLDPC codes. The main objective of this paper is to analyze the decoding failures of thehypergraph-product(HGP) andlifted-product(LP) codes and to design decoders that mitigate these failures, thus achieving a reduced error floor. The suboptimal performance of these codes can predominantly be ascribed to two structural phenomena: (1) stabilizer-induced trapping sets (TS), which correspond to stabilizer-induced subgraphs in the Tanner graphs, and (2) classical trapping sets (TS), which originate from the classical codes used in the construction of HGP and LP codes. The dynamics of stabilizer-induced TSs are examined, and a straightforward modification of iterative decoders is proposed to circumvent these TSs. Moreover, this work proposes a systematic methodology for designing decoders that can circumvent the classical TSs in both HGP and LP codes by deriving them from decoders capable of avoiding the TSs in the parent classical LDPC codes. When decoders that can avoid stabilizer-induced TSs are run in parallel with those that can mitigate the effect of classical TSs, the logical error rate improves significantly in the error-floor region.

We introduce a low-complexity message-passing quantum error correction algorithm for decoding Quantum Low-Density Parity-Check (QLDPC) stabilizer codes. The proposed decoder operates on the quaternary stabilizer graph but only exchanges binary messages. This leads to a significantly reduced complexity compared to other quaternary belief propagation (BP) algorithms that pass floating-point messages. The efficacy of the proposed decoder is evaluated by providing decoding examples, performance metrics using Monte-Carlo simulations, and complexity analysis. Despite its reduced complexity, the performance loss of the proposed decoder is modest compared to floating-point parallel quaternary decoders for a Calderbank-Shor-Steane (CSS) code family. In particular, experiments obtained over the [[1054, 140, 20]] lifted product (LP) Tanner code demonstrated that for low error rates (< 0.01), the proposed quaternary-binary message-passing decoder approaches the performance of quaternary BP by converging in almost the same number of iterations while requiring less complex operations. Additionally, for non-CSS codes, our decoder performs similarly as quaternary floating-point decoders despite its lower complexity.

This article introduces a novel communication paradigm for the unsourced, uncoordinated Gaussian multiple access problem. The major components of the envisioned framework are as follows. The encoded bits of every message are partitioned into two groups. The first portion is transmitted using a compressive sensing scheme, whereas the second set of bits is conveyed using a multi-user coding scheme. The compressive sensing portion is key in sidestepping some of the challenges posed by the unsourced aspect of the problem. The information afforded by the compressive sensing is employed to create a sparse random multi-access graph conducive to joint decoding. This construction leverages the lessons learned from traditional IDMA into creating low-complexity schemes for the unsourced setting, while also accounting for inherent randomness. Under joint message-passing decoding, the proposed scheme offers good performance at a low computational complexity. Findings are supported by numerical simulations, and results are compared to existing alternatives.

Sparse regression codes with approximate message passing (AMP) decoding have gained much attention in recent times. The concepts underlying this coding scheme extend to unsourced random access with coded compressed sensing (CCS), as first demonstrated by Fengler, Jung, and Caire. Specifically, their approach employs a concatenated coding framework with an inner AMP decoder followed by an outer tree decoder. In their original implementation, these two components work independently of each other, with the tree decoder acting on the static output of the AMP decoder. This article introduces a novel framework where the inner AMP decoder and the outer decoder operate in tandem, dynamically passing information back and forth to take full advantage of the underlying CCS structure. This scheme necessitates the redesign of the outer code as to enable belief propagation in a computationally tractable manner. The enhanced architecture exhibits significant performance benefits over a range of system parameters. The error performance of the proposed scheme can be accurately predicted through a set of equations known as state evolution of AMP. These findings are supported both analytically and through numerical methods.

This article presents a novel enhancement to the random spreading based coding scheme developed by Pradhan et al. for the unsourced multiple access channel. The original coding scheme features a polar outer code in conjunction with a successive cancellation list decoder (SCLD) and a hard-input soft-output MMSE estimator. In contrast, the proposed scheme employs a soft-input soft-output MMSE estimator for multi-user detection. This is accomplished by replacing the SCLD based polar code with an LDPC code amenable to belief propagation decoding. This novel framework is leveraged to successfully pass pertinent soft information between the MMSE estimator and the outer code. LDPC codes are carefully designed using density evolution techniques to match the iterative process. This enhanced architecture exhibits significant performance improvements and represents the state-of-the-art over a wide range of system parameters.

We consider the approximate support recovery (ASR) task of inferring the support of a$K$-sparse vector$\mathrm{x}\in \mathbb{R}^{n}$from$m$noisy measurements. We examine the case where$n$is large, which precludes the application of standard compressed sensing solvers, thereby necessitating solutions with lower complexity. We design a scheme for ASR by leveraging techniques developed for unsourced multiple access. We present two decoding algorithms with computational complexities$\mathcal{O}(K^{2}\log n+ K\log n\log\log n)$and$\mathcal{O}(K^{3}+K^{2}\log n+K\log n\log \log n)$per iteration, respectively. When$K\ll n$, this is much lower than the complexity of approximate message passing with a minimum mean squared error denoiser, which requires$\mathcal{O}(mn)$operations per iteration. This gain comes at a slight performance cost. Our findings suggest that notions from multiple access can play an important role in the design of measurement schemes for ASR.

We introduce a class of codes, called Squeezed Random Khatri-Rao Product (RKRP) codes, for coded matrix multiplication when each worker node can perform multiple submatrix products. The proposed codes are a generalization of RKRP codes in [1] and are built on the idea of squeezed polynomial codes in [2]. We show that squeezed RKRP codes are maximum distance separable with probability 1. They have the same communication cost as that of squeezed polynomial codes while offering better numerical stability.

This article presents a novel transmission scheme for the unsourced, uncoordinated Gaussian multiple access problem. The proposed scheme leverages notions from single-user coding, random spreading, minimum-mean squared error (MMSE) estimation, and successive interference cancellation. Specifically, every message is split into two parts: the first fragment serves as the argument to an injective function that determines which spreading sequence should be employed, whereas the second component of the message is encoded using a polar code. The latter coded bits are then spread using the sequence determined during the first step. The ensuing signal is transmitted through a Gaussian multiple-access channel (GMAC). On the receiver side, active sequences are detected using a correlation-based energy detector, thereby simultaneously recovering individual signature sequences and their generating information bits in the form of preimages of the sequence selection function. Using the set of detected active spreading sequences, an MMSE estimator is employed to produce log-likelihood ratios (LLRs) for the second part of the messages corresponding to these detected users. The LLRs associated with each detected user are then passed to a list decoder of the polar code, which performs single-user decoding to decode the second portion of the message. This decoding operation proceeds iteratively by subtracting the interference due to the successfully decoded messages from the received signal, and repeating the above steps on the residual signal. At this stage, the proposed algorithm outperforms alternate existing low-complexity schemes when the number of active uses is below 225.

Sparse regression codes with approximate message passing (AMP) decoding have gained much attention in recent times. The concepts underlying this coding scheme extend to unsourced access with coded compressed sensing (CCS), as first pointed out by Fengler, Jung, and Caire. More specifically, their approach uses a concatenated coding framework with an inner AMP decoder followed by an outer tree decoder. In the original implementation, these two components work independently of each other, with the tree decoder acting on the static output of the AMP decoder. This article introduces a novel framework where the inner AMP decoder and the outer tree decoder operate in tandem, dynamically passing information back and forth to take full advantage of the underlying CCS structure. The enhanced architecture exhibits significant performance benefit over a range of system parameters.

We propose two coding schemes for distributed matrix multiplication in the presence of stragglers. These coding schemes are adaptations of Luby Transform (LT) codes and Raptor codes to distributed matrix multiplication and are termedFactored LT (FLT) codesandFactored Raptor (FRT) codes. We show that all nodes in the Tanner graph of a randomly sampled code have a tree-like neighborhood with high probability. This ensures that the density evolution analysis gives a reasonable estimate of the average recovery threshold of FLT codes. The recovery threshold of the proposed FLT codes is asymptotically optimal when the output degree distribution is Soliton. Empirically, we show that FRT codes have an excellent recovery threshold while the number of worker nodes is moderately large. In addition, using Azuma–Hoeffding inequality, we derive concentration results to show that the recovery threshold of a randomly chosen FLT code is close to the ensemble average. FLT and FRT codes have better recovery thresholds when compared to Product codes and they are expected to have better numerical stability when compared to Polynomial codes, while they can also be decoded with a low-complexity decoding algorithm. Finally, the proposed codes are better matched to the practically important case of sparse matrix-matrix multiplication as compared to many previous schemes.

This work considers the distributed multivariate polynomial evaluation (DMPE) problem using a master-worker framework, which was originally considered by Yu et al., where Lagrange Coded Computing (LCC) was proposed as a coded computation scheme to provide resilience against stragglers for the DMPE problem. In this work, we propose a variant of the LCC scheme, termed Product Lagrange Coded Computing (PLCC), by combining ideas from classical product codes and LCC. The main advantage of PLCC is that they are more numerically stable than LCC; however, their resilience to stragglers is sub-optimal.

This article introduces a novel communication paradigm for the unsourced, uncoordinated Gaussian multiple access problem. The major components of the envisioned framework are as follows. The encoded bits of every message are partitioned into two groups. The first portion is transmitted using a compressive sensing scheme, whereas the second set of bits is conveyed using a multi-user coding scheme. The compressive sensing portion is key in sidestepping some of the challenges posed by the unsourced aspect of the problem. The information afforded by the compressive sensing is employed to create a sparse random multi-access graph conducive to joint decoding. This construction leverages the lessons learned from traditional IDMA into creating low- complexity schemes for the unsourced setting and its inherent randomness. Under joint message- passing decoding, the proposed scheme offers superior performance compared to existing low- complexity alternatives. Findings are supported by numerical simulations.

Block-error threshold analysis of protographs in 5G standard are considered over binary erasure channel (BEC) and binary additive white Gaussian noise (BIAWGN) channel. For protographs with degree-one variable nodes, conditions are derived to ensure equality of bit-error threshold and block-error threshold. Using this condition, it is shown that block-error threshold and bit-error threshold for protographs in in 5G standard are the same. Protographs with block-error threshold close to capacity are designed and shown to have better error rate performance than the protographs in 5G standard.

Protograph low-density-parity-check (LDPC) codes are considered to design near-capacity low-rate codes over the binary erasure channel and the binary additive white Gaussian noise channel. For protographs with degree-one variable nodes and doubly-generalized LDPC (DGLDPC) codes, conditions are derived to ensure the equality of bit-error threshold and block-error threshold. Using this condition, low-rate codes with block-error threshold close to capacity are designed and shown to have better error rate performance than other existing codes.

Protograph doubly-generalized low-density parity-check (DGLDPC) codes, which allow for arbitrary component codes at the variable and check nodes of a protograph, are considered. Exact density evolution is derived over the binary erasure channel. Conditions on the protograph and component codes to ensure equality of block-error threshold and density evolution threshold for large-girth ensembles are established. Conditions for stability of density evolution are derived, and block-error threshold property is extended to binary-input symmetric channels. Optimized low-rate protographs for DGLDPC codes over the erasure channel are presented.

Density evolution for protograph low-density parity-check (LDPC) codes is considered, and it is shown that the message-error rate falls double-exponentially with iterations whenever the degree-2 subgraph of the protograph is cycle-free and noise level is below threshold. Conditions for stability of protograph density evolution are established and related to the structure of the protograph. Using large-girth graphs, sequences of protograph LDPC codes with block-error threshold equal to bit-error threshold and block-error rate falling near-exponentially with blocklength are constructed deterministically. Small-sized protographs are optimized to obtain thresholds near capacity for binary erasure and binary-input Gaussian channels.

For certain degree-distribution pairs with non-zero fraction of degree-two bit nodes, the bit-error threshold of the standard ensemble of Low Density Parity Check (LDPC) codes is known to be close to capacity. However, the degree-two bit nodes preclude the possibility of a block-error threshold. Interestingly, LDPC codes constructed using protographs allow the possibility of having both degree-two bit nodes and a block-error threshold. In this paper, we analyze density evolution for protograph LDPC codes over the binary erasure channel and show that their bit-error probability decreases double exponentially with the number of iterations when the erasure probability is below the bit-error threshold and long chain of degree-two variable nodes are avoided in the protograph. We present deterministic constructions of such protograph LDPC codes with girth logarithmic in blocklength, resulting in an exponential fall in bit-error probability below the threshold. We provide optimized protographs, whose block-error thresholds are better than that of the standard ensemble with minimum bit-node degree three. These protograph LDPC codes are theoretically of great interest, and have applications, for instance, in coding with strong secrecy over wiretap channels.