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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. 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Based on this rich history, and by accounting for the technological advances in very large-scale integration, this article will outline some recent trends in channel coding as they may be applied in 6G systems, ranging from novel approaches for short blocklengths such as automorphism ensemble decoding, via ideas of coding for multiple access, to concepts for unified coding schemes that may simplify encoding/decoding hardware at competitive error-correcting performance.","annote":"","author":[{"family":"Miao","given":"Sisi"},{"family":"Kestel","given":"Claus"},{"family":"Johannsen","given":"Lucas"},{"family":"Geiselhart","given":"Marvin"},{"family":"Schmalen","given":"Laurent"},{"family":"Balatsoukas-Stimming","given":"Alexios"},{"family":"Liva","given":"Gianluigi"},{"family":"Wehn","given":"Norbert"},{"family":"ten Brink","given":"Stephan"}],"citation-label":"10571997","collection-editor":[],"collection-title":"","container-author":[],"container-title":"Proceedings of the IEEE","documents":[],"edition":"","editor":[],"event-date":{"date-parts":[["2024","July"]],"literal":"2024"},"event-place":"","id":"f8593f8e84508f87b4e0b31d7188a1bcmgeiselhart","interhash":"1f94fd8c86f785ad6813d931d681dd81","intrahash":"f8593f8e84508f87b4e0b31d7188a1bc","issue":"7","issued":{"date-parts":[["2024","July"]],"literal":"2024"},"keyword":"coding myown","misc":{"issn":"1558-2256","doi":"10.1109/JPROC.2024.3416050"},"note":"","number":"7","number-of-pages":"22","page":"653-675","page-first":"653","publisher":"","publisher-place":"","status":"","title":"Trends in Channel Coding for 6G","type":"article-journal","username":"mgeiselhart","version":"","volume":"112"},"f8593f8e84508f87b4e0b31d7188a1bcinue":{"DOI":"10.1109/JPROC.2024.3416050","ISBN":"","ISSN":"1558-2256","URL":"","abstract":"Error correction coding (i.e., channel coding) is a key ingredient of any digital communications system. 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Based on this rich history, and by accounting for the technological advances in very large-scale integration, this article will outline some recent trends in channel coding as they may be applied in 6G systems, ranging from novel approaches for short blocklengths such as automorphism ensemble decoding, via ideas of coding for multiple access, to concepts for unified coding schemes that may simplify encoding/decoding hardware at competitive error-correcting performance.","annote":"","author":[{"family":"Miao","given":"Sisi"},{"family":"Kestel","given":"Claus"},{"family":"Johannsen","given":"Lucas"},{"family":"Geiselhart","given":"Marvin"},{"family":"Schmalen","given":"Laurent"},{"family":"Balatsoukas-Stimming","given":"Alexios"},{"family":"Liva","given":"Gianluigi"},{"family":"Wehn","given":"Norbert"},{"family":"ten Brink","given":"Stephan"}],"citation-label":"10571997","collection-editor":[],"collection-title":"","container-author":[],"container-title":"Proceedings of the IEEE","documents":[],"edition":"","editor":[],"event-date":{"date-parts":[["2024","July"]],"literal":"2024"},"event-place":"","id":"f8593f8e84508f87b4e0b31d7188a1bcinue","interhash":"1f94fd8c86f785ad6813d931d681dd81","intrahash":"f8593f8e84508f87b4e0b31d7188a1bc","issue":"7","issued":{"date-parts":[["2024","July"]],"literal":"2024"},"keyword":"myown coding","misc":{"issn":"1558-2256","doi":"10.1109/JPROC.2024.3416050"},"note":"","number":"7","number-of-pages":"22","page":"653-675","page-first":"653","publisher":"","publisher-place":"","status":"","title":"Trends in Channel Coding for 6G","type":"article-journal","username":"inue","version":"","volume":"112"},"ae723295e9ffcc9596d5a938a450da85inue":{"DOI":"10.1109/IEEECONF62907.2025.10949101","ISBN":"","ISSN":"","URL":"","abstract":"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 code-word 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.","annote":"","author":[{"family":"Krieg","given":"Felix"},{"family":"Clausius","given":"Jannis"},{"family":"Rübenacke","given":"Marvin"},{"family":"ten Brink","given":"Stephan"}],"citation-label":"10949101","collection-editor":[],"collection-title":"","container-author":[],"container-title":"2025 14th International ITG Conference on Systems, Communications and Coding (SCC)","documents":[],"edition":"","editor":[],"event-date":{"date-parts":[["2025","March"]],"literal":"2025"},"event-place":"","id":"ae723295e9ffcc9596d5a938a450da85inue","interhash":"37880ae95bb60a15d70bd7cb9d3a85d0","intrahash":"ae723295e9ffcc9596d5a938a450da85","issue":"","issued":{"date-parts":[["2025","March"]],"literal":"2025"},"keyword":"myown coding","misc":{"doi":"10.1109/IEEECONF62907.2025.10949101"},"note":"","number":"","number-of-pages":"5","page":"1-6","page-first":"1","publisher":"","publisher-place":"","status":"","title":"A Comparative Study of Ensemble Decoding Methods for Short Length LDPC Codes","type":"paper-conference","username":"inue","version":"","volume":""},"ae723295e9ffcc9596d5a938a450da85mgeiselhart":{"DOI":"10.1109/IEEECONF62907.2025.10949101","ISBN":"","ISSN":"","URL":"","abstract":"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 code-word 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.","annote":"","author":[{"family":"Krieg","given":"Felix"},{"family":"Clausius","given":"Jannis"},{"family":"Rübenacke","given":"Marvin"},{"family":"ten Brink","given":"Stephan"}],"citation-label":"10949101","collection-editor":[],"collection-title":"","container-author":[],"container-title":"2025 14th International ITG Conference on Systems, Communications and Coding (SCC)","documents":[],"edition":"","editor":[],"event-date":{"date-parts":[["2025","March"]],"literal":"2025"},"event-place":"","id":"ae723295e9ffcc9596d5a938a450da85mgeiselhart","interhash":"37880ae95bb60a15d70bd7cb9d3a85d0","intrahash":"ae723295e9ffcc9596d5a938a450da85","issue":"","issued":{"date-parts":[["2025","March"]],"literal":"2025"},"keyword":"coding myown","misc":{"doi":"10.1109/IEEECONF62907.2025.10949101"},"note":"","number":"","number-of-pages":"5","page":"1-6","page-first":"1","publisher":"","publisher-place":"","status":"","title":"A Comparative Study of Ensemble Decoding Methods for Short Length LDPC Codes","type":"paper-conference","username":"mgeiselhart","version":"","volume":""},"431a23b580b6f34091d0c209bf2168dbmgeiselhart":{"DOI":"10.1109/TCOMM.2024.3432688","ISBN":"","ISSN":"1558-0857","URL":"","abstract":"Row-merged polar codes are a family of pre-transformed polar codes (PTPCs) with little precoding overhead. Providing an improved distance spectrum over plain polar codes, they are capable to perform close to the finite-length capacity bounds. However, there is still a lack of efficient design procedures for row-merged polar codes. Using novel weight enumeration algorithms with low computational complexity, we propose a design methodology for row-merged polar codes that directly considers their minimum distance properties. The codes significantly outperform state-of-the-art cyclic redundancy check (CRC)-aided polar codes under successive cancellation list (SCL) decoding in error-correction performance. Furthermore, we present fast simplified successive cancellation list (Fast-SSCL) decoding of PTPCs, based on which we derive a high-throughput, unrolled architecture template for fully pipelined decoders. Implementation results of SCL decoders for row-merged polar codes in a 12nm technology additionally demonstrate the superiority of these codes with respect to the implementation costs, compared to state-of-the-art reference decoder implementations.","annote":"","author":[{"family":"Zunker","given":"Andreas"},{"family":"Geiselhart","given":"Marvin"},{"family":"Johannsen","given":"Lucas"},{"family":"Kestel","given":"Claus"},{"family":"ten Brink","given":"Stephan"},{"family":"Vogt","given":"Timo"},{"family":"Wehn","given":"Norbert"}],"citation-label":"10608040","collection-editor":[],"collection-title":"","container-author":[],"container-title":"IEEE Transactions on Communications","documents":[],"edition":"","editor":[],"event-date":{"date-parts":[["2025","jan"]],"literal":"2025"},"event-place":"","id":"431a23b580b6f34091d0c209bf2168dbmgeiselhart","interhash":"9a67c8ffc810af63096c674f85ed70ad","intrahash":"431a23b580b6f34091d0c209bf2168db","issue":"1","issued":{"date-parts":[["2025","jan"]],"literal":"2025"},"keyword":"coding myown","misc":{"issn":"1558-0857","doi":"10.1109/TCOMM.2024.3432688"},"note":"","number":"1","number-of-pages":"14","page":"39-53","page-first":"39","publisher":"","publisher-place":"","status":"","title":"Row-Merged Polar Codes: Analysis, Design, and Decoder Implementation","type":"article-journal","username":"mgeiselhart","version":"","volume":"73"},"431a23b580b6f34091d0c209bf2168dbinue":{"DOI":"10.1109/TCOMM.2024.3432688","ISBN":"","ISSN":"1558-0857","URL":"","abstract":"Row-merged polar codes are a family of pre-transformed polar codes (PTPCs) with little precoding overhead. Providing an improved distance spectrum over plain polar codes, they are capable to perform close to the finite-length capacity bounds. However, there is still a lack of efficient design procedures for row-merged polar codes. Using novel weight enumeration algorithms with low computational complexity, we propose a design methodology for row-merged polar codes that directly considers their minimum distance properties. The codes significantly outperform state-of-the-art cyclic redundancy check (CRC)-aided polar codes under successive cancellation list (SCL) decoding in error-correction performance. Furthermore, we present fast simplified successive cancellation list (Fast-SSCL) decoding of PTPCs, based on which we derive a high-throughput, unrolled architecture template for fully pipelined decoders. Implementation results of SCL decoders for row-merged polar codes in a 12nm technology additionally demonstrate the superiority of these codes with respect to the implementation costs, compared to state-of-the-art reference decoder implementations.","annote":"","author":[{"family":"Zunker","given":"Andreas"},{"family":"Geiselhart","given":"Marvin"},{"family":"Johannsen","given":"Lucas"},{"family":"Kestel","given":"Claus"},{"family":"ten Brink","given":"Stephan"},{"family":"Vogt","given":"Timo"},{"family":"Wehn","given":"Norbert"}],"citation-label":"10608040","collection-editor":[],"collection-title":"","container-author":[],"container-title":"IEEE Transactions on Communications","documents":[],"edition":"","editor":[],"event-date":{"date-parts":[["2025","jan"]],"literal":"2025"},"event-place":"","id":"431a23b580b6f34091d0c209bf2168dbinue","interhash":"9a67c8ffc810af63096c674f85ed70ad","intrahash":"431a23b580b6f34091d0c209bf2168db","issue":"1","issued":{"date-parts":[["2025","jan"]],"literal":"2025"},"keyword":"myown coding","misc":{"issn":"1558-0857","doi":"10.1109/TCOMM.2024.3432688"},"note":"","number":"1","number-of-pages":"14","page":"39-53","page-first":"39","publisher":"","publisher-place":"","status":"","title":"Row-Merged Polar Codes: Analysis, Design, and Decoder Implementation","type":"article-journal","username":"inue","version":"","volume":"73"},"501a80ae3d850713ecb94bb36af28d00mgeiselhart":{"DOI":"10.1109/IEEECONF56349.2022.10051937","ISBN":"","ISSN":"2576-2303","URL":"","abstract":"It is well known that to fulfill their full potential, the design of polar codes must be tailored to their intended decoding algorithm. While for successive cancellation (SC) decoding, information theoretically optimal constructions are available, the code design for other decoding algorithms (such as belief propagation (BP) decoding) can only be optimized using extensive Monte Carlo simulations. We propose to view the design process of polar codes as a graph search problem and thereby approaching it more systematically. Based on this formalism, the design-time complexity can be significantly reduced compared to state-of-the-art Genetic Algorithm (GenAlg) and deep learning-based design algorithms. Moreover, sequences of rate-compatible polar codes can be efficiently found. Finally, we analyze both the complexity of the proposed algorithm and the error-rate performance of the constructed codes.","annote":"","author":[{"family":"Geiselhart","given":"Marvin"},{"family":"Zunker","given":"Andreas"},{"family":"Elkelesh","given":"Ahmed"},{"family":"Clausius","given":"Jannis"},{"family":"Brink","given":"Stephan ten"}],"citation-label":"10051937","collection-editor":[],"collection-title":"","container-author":[],"container-title":"2022 56th Asilomar Conference on Signals, Systems, and Computers","documents":[],"edition":"","editor":[],"event-date":{"date-parts":[["2022","oct"]],"literal":"2022"},"event-place":"","id":"501a80ae3d850713ecb94bb36af28d00mgeiselhart","interhash":"777392a621a63a1904c357874e242839","intrahash":"501a80ae3d850713ecb94bb36af28d00","issue":"","issued":{"date-parts":[["2022","oct"]],"literal":"2022"},"keyword":"coding myown","misc":{"issn":"2576-2303","doi":"10.1109/IEEECONF56349.2022.10051937"},"note":"","number":"","number-of-pages":"4","page":"387-391","page-first":"387","publisher":"","publisher-place":"","status":"","title":"Graph Search based Polar Code Design","type":"paper-conference","username":"mgeiselhart","version":"","volume":""},"501a80ae3d850713ecb94bb36af28d00inue":{"DOI":"10.1109/IEEECONF56349.2022.10051937","ISBN":"","ISSN":"2576-2303","URL":"","abstract":"It is well known that to fulfill their full potential, the design of polar codes must be tailored to their intended decoding algorithm. While for successive cancellation (SC) decoding, information theoretically optimal constructions are available, the code design for other decoding algorithms (such as belief propagation (BP) decoding) can only be optimized using extensive Monte Carlo simulations. We propose to view the design process of polar codes as a graph search problem and thereby approaching it more systematically. Based on this formalism, the design-time complexity can be significantly reduced compared to state-of-the-art Genetic Algorithm (GenAlg) and deep learning-based design algorithms. Moreover, sequences of rate-compatible polar codes can be efficiently found. Finally, we analyze both the complexity of the proposed algorithm and the error-rate performance of the constructed codes.","annote":"","author":[{"family":"Geiselhart","given":"Marvin"},{"family":"Zunker","given":"Andreas"},{"family":"Elkelesh","given":"Ahmed"},{"family":"Clausius","given":"Jannis"},{"family":"Brink","given":"Stephan ten"}],"citation-label":"10051937","collection-editor":[],"collection-title":"","container-author":[],"container-title":"2022 56th Asilomar Conference on Signals, Systems, and Computers","documents":[],"edition":"","editor":[],"event-date":{"date-parts":[["2022","oct"]],"literal":"2022"},"event-place":"","id":"501a80ae3d850713ecb94bb36af28d00inue","interhash":"777392a621a63a1904c357874e242839","intrahash":"501a80ae3d850713ecb94bb36af28d00","issue":"","issued":{"date-parts":[["2022","oct"]],"literal":"2022"},"keyword":"myown from:mgeiselhart coding","misc":{"issn":"2576-2303","doi":"10.1109/IEEECONF56349.2022.10051937"},"note":"","number":"","number-of-pages":"4","page":"387-391","page-first":"387","publisher":"","publisher-place":"","status":"","title":"Graph Search based Polar Code Design","type":"paper-conference","username":"inue","version":"","volume":""}}