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.

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.

Auxiliary Nodes for BP Decoding of Quantum LDPC Codes

Many recently proposed Calderbank-Shor-Steane (CSS) quantum low-density parity-check (QLDPC) codes have sparse decoding graphs, enabling syndrome-based belief propagation (BP) decoding at low complexity. Their construction, however, often results in properties that impair BP performance, such as short cycles and degeneracy. In this work, we propose a general framework for introducing auxiliary variable nodes (AVNs) and auxiliary check nodes (ACNs) into the decoding graph of CSS codes, compatible with the standard stabilizer measurement framework. This provides an additional degree of freedom in the design of the decoding graph itself and can be used to tackle the aforementioned shortcomings. We show that recently proposed techniques, 4-cycle removal and subcode ensemble decoding, can be interpreted as instances of this framework. For 4-cycle removal, we find that the gains depend strongly on the BP iteration count and check-node message scaling. Building on this framework, we further propose a graph-derived subcode ensemble decoder and demonstrate under circuit-level noise that it substantially reduces the per-round logical error rate compared with BP on the corresponding 4-cycle-free decoding graph.

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.