arXiv subjects

Felix Krieg

Publications and source records attributed to Felix Krieg.

7 recordsLinked to original sources

arXiv2026

Row-Boosted Ensemble Belief Propagation for Short LDPC Codes

Belief-propagation (BP) decoding of short and moderate-length low-density parity-check (LDPC) codes is limited by finite-length graph effects: a single decoder trajectory can become trapped or oscillatory even when an alternative trajectory would decode the received word. Existing ensemble-BP decoders create the required diversity through multiple parity-check matrices, automorphisms, modified schedules, subcodes, or altered update rules. We introduce row-boosted ensemble (RBE) decoding as a minimal decoder-side diversity mechanism: all ensemble members share the same parity-check matrix and the same BP kernel, and differ only in a small set of parity-check rows whose outgoing messages are boosted. On the 5G~NR BG1 $(144,96)$ code, RBE with $32$ members lowers the frame-error rate of BP with $20$ iterations (BP-20) from $1.6\times10^{-2}$ to $1.6\times10^{-3}$ at $E_\mathrm{b}/N_0=4.0\,\mathrm{dB}$, outperforming saturated-min-sum and affine subcode ensembles of equal size. Increasing the ensemble size yields additional gains, indicating that RBE provides a scalable performance-complexity tradeoff. The gains transfer across 5G~NR block lengths and rates, to non-5G short LDPC codes, and across flooding and layered schedules.

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arXiv2026

SECS: A Soft Ensemble-Combining Stage for Low-Latency Decoding

Ensemble decoding is a promising technique for ultra-reliable low-latency communication, as it trades hardware parallelism for decoding latency by running M diverse belief propagation (BP) decoders in parallel. Conventional ensembles select their output from the candidates. Hence, decoding fails whenever no member finds the correct codeword. In this work, we show that shallow, diverse BP members are poor correctors, but excellent sorters. Based on this observation, we propose a soft ensemble-combining stage (SECS) that combines extrinsic messages after only a few iterations, yielding a reliability ordering whose most reliable positions are nearly error-free. A subsequent re-encoding stage (like ordered-statistics decoding (OSD)) converts this ordering into a near-maximum likelihood (ML) candidate codeword, at a fraction of the BP latency of a fully converged ensemble. We demonstrate the proposed scheme on three short codes: a (63,30) Bose-Ray-Chaudhuri-Hocquenghem code, an overcomplete PG(2,8) code, and a search-designed (105,53) cyclic code. In all cases, the SECS with OSD post-processing closes most of the gap to ML decoding, while significantly reducing the number of required BP iterations.

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arXiv2026

Towards a Unified Coding Scheme for 6G

The growing demand for higher data rates necessitates continuous innovations in wireless communication systems, particularly with the emergence of 6G. Channel coding plays a crucial role in this evolution. In 5G systems, rate-adaptive raptor-like quasi-cyclic irregular low-density parity-check codes are used for the data link, while polar codes with successive cancellation list decoding handle short messages on the synchronization channel. However, to meet the stringent requirements of future 6G systems, a versatile and unified coding scheme should be developed - one that offers competitive error-correcting performance alongside low complexity encoding and decoding schemes that enable energy-efficient hardware implementations. This white paper outlines the vision for such a unified coding scheme. We explore various 6G communication scenarios that pose new challenges to channel coding and provide a first analysis of potential solutions.

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arXiv2025

Long Polar vs. LDPC Codes under Complexity-Constrained Decoding

The prevailing opinion in industry and academia is that polar codes are competitive for short code lengths, but can no longer keep up with low-density parity-check (LDPC) codes as block length increases. This view is typically based on the assumption that LDPC codes can be decoded with a large number of belief propagation (BP) iterations. However, in practice, the number of iterations may be rather limited due to latency and complexity constraints. In this paper, we show that for a similar number of fixed-point log-likelihood ratio (LLR) operations, long polar codes under successive cancellation (SC) decoding outperform their LDPC counterparts. In particular, simplified successive cancellation (SSC) decoding of polar codes exhibits a better complexity scaling than $N \log{N}$ and requires fewer operations than a single BP iteration of an LDPC code with the same parameters.

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arXiv2024

A Comparative Study of Ensemble Decoding Methods for Short Length LDPC Codes

To alleviate the suboptimal performance of belief propagation (BP) decoding of short low-density parity-check (LDPC) codes, a plethora of improved decoding algorithms has been proposed over the last two decades. Many of these methods can be described using the same general framework, which we call ensemble decoding: A set of independent constituent decoders works in parallel on the received sequence, each proposing a codeword candidate. From this list, the maximum likelihood (ML) decision is designated as the decoder output. In this paper, we qualitatively and quantitatively compare different realizations of the ensemble decoder, namely multiple-bases belief propagation (MBBP), automorphism ensemble decoding (AED), scheduling ensemble decoding (SED), noise-aided ensemble decoding (NED) and saturated belief propagation (SBP). While all algorithms can provide gains over traditional BP decoding, ensemble methods that exploit the code structure, such as MBBP and AED, typically show greater performance improvements.

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arXiv2024

Nested Symmetric Polar Codes

In this paper, we propose a data-driven algorithm to design rate- and length-flexible polar codes. While the algorithm is very general, a particularly appealing use case is the design of codes for automorphism ensemble decoding (AED), a promising decoding algorithm for ultra-reliable low-latency communications (URLLC) and massive machine-type communications (mMTC) applications. To this end, theoretic results on nesting of symmetric polar codes are derived, which give hope in finding a fully nested, rate-compatible sequence suitable for AED. Using the proposed algorithms, such a flexible polar code design for automorphism ensemble successive cancellation (SC) decoding is constructed, outperforming existing code designs for AED and also the 5G polar code under cyclic redundancy check (CRC)-aided successive cancellation list (SCL) decoding.

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arXiv2023

Phase-Equivariant Polar Coded Modulation

For short-packet, low-latency communications over random access channels, piloting overhead significantly reduces spectral efficiency. Therefore, pilotless systems recently gained attraction. While blind phase estimation algorithms such as Viterbi-Viterbi Phase Estimation (VVPE) can correct a phase offset using only payload symbols, a phase ambiguity remains. We first show that the remaining phase rotations in a polar coded quadrature amplitude modulation (QAM) transmission with gray labeling are combinations of bit-flips and automorphisms. Therefore, the decoder is equivariant to such phase rotations and, by smartly selecting the frozen bits, one can jointly decode and resolve the phase ambiguity, without the need for pilot symbols or an outer code. Our proposed system outperforms pilot-assisted transmissions by up to 0.8 dB and 2 dB for quaternary phase shift keying (QPSK) and 16-QAM, respectively.

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