arXiv subjects

Holger Jäkel

Publications and source records attributed to Holger Jäkel.

17 recordsLinked to original sources

arXiv2026

Affine Subcode Ensemble Decoding for Degeneracy-Aware Quantum Error Correction

Quantum low-density parity-check codes are promising candidates for low-overhead fault-tolerant quantum computing, but degeneracy is known to impair the convergence of belief-propagation (BP) decoding of these codes. In this work, we show that appending linearly independent rows to a check matrix of a stabilizer code can reduce the search space for a valid degenerate solution. Motivated by this, we extend the recently proposed affine subcode ensemble decoding technique from the classical to the quantum setting. Moreover, we employ overcomplete matrices for each decoding path. Monte-Carlo simulations on toric and generalized bicycle codes demonstrate improved convergence and reduced logical error rate.

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arXiv2026

Affine Subcode Ensemble Decoding of Linear Block Codes

In the short block length regime, ensemble decoding schemes with their inherently parallel structure can improve error correction performance and reduce latency compared to stand-alone suboptimal decoders such as belief propagation (BP). In this work, we introduce affine subcode ensemble decoding (aSCED), which uses an ensemble of decoders operating on linear block codes and both linear and strictly affine subcodes. This generalizes the recently proposed subcode ensemble decoding (SCED), which is restricted to linear subcodes. We derive BP update rules for affine subcodes and show that aSCED simplifies ensemble design compared to SCED, multiple bases BP, and automorphism ensemble decoding. Monte-Carlo simulations of two low-density parity-check codes and two Bose-Chaudhuri-Hocquenghem (BCH) codes demonstrate improved error correction performance of aSCED over competing existing ensemble schemes. Notably, for one BCH code, when combining ensemble design with algorithms for constructing high-performance parity-check matrices, aSCED achieves near-maximum likelihood performance using only 64 BP decoding paths.

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arXiv2026

On the Error Rate of Binary BCH Codes under Error-and-erasure Decoding

Determining the exact decoding error probability of linear block codes is an interesting problem. For binary BCH codes, McEliece derived methods to estimate the error probability of a simple bounded distance decoding (BDD) for BCH codes. However, BDD falls short in many applications. In this work, we consider error-and-erasure decoding and its improved variants. We derive closed-form expressions for their error probabilities and validate them through simulations. Then, we illustrate their use in assessing concatenated coding schemes.

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arXiv2025

Subcode Ensemble Decoding of Polar Codes

In the short block length regime, pre-transformed polar codes together with successive cancellation list (SCL) decoding possess excellent error correction capabilities. However, in practice, the list size is limited due to the suboptimal scaling of the required area in hardware implementations. Automorphism ensemble decoding (AED) can improve performance for a fixed list size by running multiple parallel SCL decodings on permuted received words, yielding a list of estimates from which the final estimate is selected. Yet, AED is limited to appropriately designed polar codes. Subcode ensemble decoding (ScED) was recently proposed for low-density parity-check codes and does not impose such design constraints. It uses multiple decodings in different subcodes, ensuring that the selected subcodes jointly cover the original code. We extend ScED to polar codes by expressing polar subcodes through suitable pre-transformations (PTs). To this end, we describe a framework classifying pre-transformations for pre-transformed polar codes based on their role in encoding and decoding. Within this framework, we propose a new type of PT enabling ScED for polar codes, analyze its properties, and discuss how to construct an efficient ensemble.

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arXiv2025

Removal of Small Weight Stopping Sets for Asynchronous Unsourced Multiple Access

In this paper, we analyze the formation of small stopping sets in joint factor graphs describing a frame-asynchronous two-user transmission. Furthermore, we propose an algorithm to completely avoid small stopping sets in the joint factor graph over the entire range of symbol delays. The error floor caused by these stopping sets is completely mitigated. Our key observation is that, while the order of bits in the codeword is irrelevant in a single-user environment, it turns out to be crucial in an asynchronous, unsourced two-user system. Subsequently, our algorithm finds a reordering of variable nodes which avoids the smallest stopping set in the joint graph. We show that further improvements can be achieved when girth optimization of the single-user graphs by progressive edge growth (PEG) is used in combination with our proposed algorithm. Starting with a randomized code construction with optimized degree distribution, our simulation results show that PEG followed by the proposed algorithm can improve the average per user probability of error in a noiseless channel by almost two orders of magnitude for a broad range of frame delays.

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arXiv2025

Subcode Ensemble Decoding of Linear Block Codes

Low-density parity-check (LDPC) codes together with belief propagation (BP) decoding yield exceptional error correction capabilities in the large block length regime. Yet, there remains a gap between BP decoding and maximum likelihood decoding for short block length LDPC codes. In this context, ensemble decoding schemes yield both reduced latency and good error rates. In this paper, we propose subcode ensemble decoding (SCED), which employs an ensemble of decodings on different subcodes of the code. To ensure that all codewords are decodable, we use the concept of linear coverings and explore approaches for sampling suitable ensembles for short block length LDPC codes. Monte-Carlo simulations conducted for three LDPC codes demonstrate that SCED improves decoding performance compared to stand-alone decoding and automorphism ensemble decoding. In particular, in contrast to existing schemes, e.g., multiple bases belief propagation and automorphism ensemble decoding, SCED does not require the NP-complete search for low-weight dual codewords or knowledge of the automorphism group of the code, which is often unknown.

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arXiv2024

List-based Optimization of Proximal Decoding for LDPC Codes

In this paper, the proximal decoding algorithm is considered within the context of additive white Gaussian noise (AWGN) channels. An analysis of the convergence behavior of the algorithm shows that proximal decoding inherently enters an oscillating behavior of the estimate after a certain number of iterations. Due to this oscillation, frame errors arising during decoding can often be attributed to only a few remaining wrongly decoded bit positions. In this letter, an improvement of the proximal decoding algorithm is proposed by establishing an additional step, in which these erroneous positions are attempted to be corrected. We suggest an empirical rule with which the components most likely needing correction can be determined. Using this insight and performing a subsequent ``ML-in-the-list'' decoding, a gain of up to 1 dB is achieved compared to conventional proximal decoding, depending on the decoder parameters and the code.

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arXiv2024

Improved Generalized Automorphism Belief Propagation Decoding

With the increasing demands on future wireless systems, new design objectives become eminent. Low-density parity-check codes together with belief propagation (BP) decoding have outstanding performance for large block lengths. Yet, for future wireless systems, good decoding performance for short block lengths is mandatory, a regime in which BP decoding typically shows a significant gap to maximum likelihood decoding. Automorphism ensemble decoding (AED) is known to reduce this gap effectively and, in addition, enables an easy trade-off between latency, throughput, and complexity. Recently, generalized AED (GAED) was proposed to increase the set of feasible automorphisms suitable for ensemble decoding. By construction, GAED requires a preprocessing step within its constituent paths that results in information loss and potentially limits the gains of GAED. In this work, we show that the preprocessing step can be merged with the Tanner graph of BP decoding, thereby improving the performance of the constituent paths. Finally, we show that the improvement of the individual paths also enhances the overall performance of the ensemble.

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arXiv2024

A Joint Code and Belief Propagation Decoder Design for Quantum LDPC Codes

Quantum low-density parity-check (QLDPC) codes are among the most promising candidates for future quantum error correction schemes. However, a limited number of short to moderate-length QLDPC codes have been designed and their decoding performance is sub-optimal with a quaternary belief propagation (BP) decoder due to unavoidable short cycles in their Tanner graphs. In this paper, we propose a novel joint code and decoder design for QLDPC codes. The constructed codes have a minimum distance of about the square root of the block length. In addition, it is, to the best of our knowledge, the first QLDPC code family where BP decoding is not impaired by short cycles of length 4. This is achieved by using an ensemble BP decoder mitigating the influence of assembled short cycles. We outline two code construction methods based on classical quasi-cyclic codes and finite geometry codes. Numerical results demonstrate outstanding decoding performance over depolarizing channels.

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arXiv2024

Performance Analysis of Generalized Product Codes with Irregular Degree Distribution

This paper investigates the theoretical analysis of intrinsic message passing decoding for generalized product codes (GPCs) with irregular degree distributions, a generalization of product codes that allows every code bit to be protected by a minimum of two and potentially more component codes. We derive a random hypergraph-based asymptotic performance analysis for GPCs, extending previous work that considered the case where every bit is protected by exactly two component codes. The analysis offers a new tool to guide the code design of GPCs by providing insights into the influence of degree distributions on the performance of GPCs.

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arXiv2024

Endomorphisms of Linear Block Codes

The automorphism groups of various linear codes are extensively studied yielding insights into the respective code structure. This knowledge is used in, e.g., theoretical analysis and in improving decoding performance, motivating the analyses of endomorphisms of linear codes. In this work, we discuss the structure of the set of transformation matrices of code endomorphisms, defined as a generalization of code automorphisms, and provide an explicit construction of a bijective mapping between the image of an endomorphism and its canonical quotient space. Furthermore, we introduce a one-to-one mapping between the set of transformation matrices of endomorphisms and a larger linear block code enabling the use of well-known algorithms for the search for suitable endomorphisms. Additionally, we propose an approach to obtain unknown code endomorphisms based on automorphisms of the code. Furthermore, we consider ensemble decoding as a possible use case for endomorphisms by introducing endomorphism ensemble decoding. Interestingly, EED can improve decoding performance when other ensemble decoding schemes are not applicable.

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arXiv2023

Generalized Automorphisms of Channel Codes: Properties, Code Design, and a Decoder

Low-density parity-check codes together with belief propagation (BP) decoding are known to be well-performing for large block lengths. However, for short block lengths there is still a considerable gap between the performance of the BP decoder and the maximum likelihood decoder. Different ensemble decoding schemes such as, e.g., the automorphism ensemble decoder (AED), can reduce this gap in short block length regime. We propose a generalized AED (GAED) that uses automorphisms according to the definition in linear algebra. Here, an automorphism of a vector space is defined as a linear, bijective self-mapping, whereas in coding theory self-mappings that are scaled permutations are commonly used. We show that the more general definition leads to an explicit joint construction of codes and automorphisms, and significantly enlarges the search space for automorphisms of existing linear codes. Furthermore, we prove the concept that generalized automorphisms can indeed be used to improve decoding. Additionally, we propose a code construction of parity check codes enabling the construction of codes with suitably designed automorphisms. Finally, we analyze the decoding performances of the GAED for some of our constructed codes.

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arXiv2022

Semi-Deterministic Subspace Selection for Sparse Recursive Projection-Aggregation Decoding of Reed-Muller Codes

Recursive projection aggregation (RPA) decoding as introduced in [1] is a novel decoding algorithm which performs close to the maximum likelihood decoder for short-length Reed-Muller codes. Recently, an extension to RPA decoding, called sparse multi-decoder RPA (SRPA), has been proposed [2]. The SRPA approach makes use of multiple pruned RPA decoders to lower the amount of computations while keeping the performance loss small compared to RPA decoding. However, the use of multiple sparse decoders again increases the computational burden. Therefore, the focus is on the optimization of sparse single-decoder RPA decoding to keep the complexity small. In this paper, a novel method is proposed, to select subsets of subspaces used in the projection and aggregation step of SRPA decoding in order to decrease the decoding error probability on AWGN channels. The proposed method replaces the random selection of subspace subsets with a semi-deterministic selection method based on a figure of merit that evaluates the performance of each subspace. Our simulation results show that the semi-deterministic subspace selection improves the decoding performance up to $0.2\,\text{dB}$ compared to SRPA. At the same time, the complexity of SRPA decoding for RM codes of order $r\geq 3$ is reduced by up to 81% compared to SRPA.

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arXiv2014

Interference in Poisson Networks with Isotropically Distributed Nodes

Practical wireless networks are finite, and hence non-stationary with nodes typically non-homo-geneously deployed over the area. This leads to a location-dependent performance and to boundary effects which are both often neglected in network modeling. In this work, interference in networks with nodes distributed according to an isotropic but not necessarily stationary Poisson point process (PPP) are studied. The resulting link performance is precisely characterized as a function of (i) an arbitrary receiver location and of (ii) an arbitrary isotropic shape of the spatial distribution. Closed-form expressions for the first moment and the Laplace transform of the interference are derived for the path loss exponents $α=2$ and $α=4$, and simple bounds are derived for other cases. The developed model is applied to practical problems in network analysis: for instance, the accuracy loss due to neglecting border effects is shown to be undesirably high within transition regions of certain deployment scenarios. Using a throughput metric not relying on the stationarity of the spatial node distribution, the spatial throughput locally around a given node is characterized.

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arXiv2013

Cooperative Relaying in a Poisson Field of Interferers: A Diversity Order Analysis

This work analyzes the gains of cooperative relaying in interference-limited networks, in which outages can be due to interference and fading. A stochastic model based on point process theory is used to capture the spatial randomness present in contemporary wireless networks. Using a modification of the diversity order metric, the reliability gain of selection decode-and-forward is studied for several cases. The main results are as follows: the achievable \emph{spatial-contention} diversity order (SC-DO) is equal to one irrespective of the type of channel which is due to the ineffectiveness of the relay in the MAC-phase (transmit diversity). In the BC-phase (receive diversity), the SC-DO depends on the amount of fading and spatial interference correlation. In the absence of fading, there is a hard transition between SC-DO of either one or two, depending on the system parameters.

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arXiv2012

Interference and Throughput in Aloha-based Ad Hoc Networks with Isotropic Node Distribution

We study the interference and outage statistics in a slotted Aloha ad hoc network, where the spatial distribution of nodes is non-stationary and isotropic. In such a network, outage probability and local throughput depend on both the particular location in the network and the shape of the spatial distribution. We derive in closed-form certain distributional properties of the interference that are important for analyzing wireless networks as a function of the location and the spatial shape. Our results focus on path loss exponents 2 and 4, the former case not being analyzable before due to the stationarity assumption of the spatial node distribution. We propose two metrics for measuring local throughput in non-stationary networks and discuss how our findings can be applied to both analysis and optimization.

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arXiv2010

Resolving the Connectivity-Throughput Trade-Off in Random Networks

The discrepancy between the upper bound on throughput in wireless networks and the throughput scaling in random networks which is also known as the connectivity-throughput trade-off is analyzed. In a random network with $λ$ nodes per unit area, throughput is found to scale by a factor of $\sqrt{\logλ}$ worse compared to the upper bound which is due to the uncertainty in the nodes' location. In the present model, nodes are assumed to know their geographical location and to employ power control, which we understand as an additional degree of freedom to improve network performance. The expected throughput-progress and the expected packet delay normalized to the one-hop progress are chosen as performance metrics. These metrics are investigated for a nearest neighbor forwarding strategy, which benefits from power control by reducing transmission power and, hence spatial contention. It is shown that the connectivity-throughput trade-off can be resolved if nodes employ a nearest neighbor forwarding strategy, achieving the upper bound on throughput on average also in a random network while ensuring asymptotic connectivity. In this case, the optimal throughput-delay scaling trade-off is also achieved.

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