{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T17:12:42Z","timestamp":1774631562406,"version":"3.50.1"},"reference-count":50,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2022,6,3]],"date-time":"2022-06-03T00:00:00Z","timestamp":1654214400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"JITSF","award":["jit-b-202110"],"award-info":[{"award-number":["jit-b-202110"]}]},{"name":"JITSF","award":["61871376"],"award-info":[{"award-number":["61871376"]}]},{"DOI":"10.13039\/501100001809","name":"NSFC","doi-asserted-by":"publisher","award":["jit-b-202110"],"award-info":[{"award-number":["jit-b-202110"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"NSFC","doi-asserted-by":"publisher","award":["61871376"],"award-info":[{"award-number":["61871376"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper presents an online perturbed and directed neural-evolutionary (Online-PDNE) decoding algorithm for polar codes, in which the perturbation noise and online directed neuro-evolutionary noise sequences are sequentially added to the received sequence for re-decoding if the standard polar decoding fails. The new decoding algorithm converts uncorrectable received sequences into error-correcting regions of their decoding space for correct decoding by adding specific noises. To reduce the decoding complexity and delay, the PDNE decoding algorithm and sole neural-evolutionary (SNE) decoding algorithm for polar codes are further proposed, which provide a considerable tradeoff between the decoding performance and complexity by acquiring the neural-evolutionary noise in an offline manner. Numerical results suggest that our proposed decoding algorithms outperform the other conventional decoding algorithms. At high signal-to-noise ratio (SNR) region, the Online-PDNE decoding algorithm improves bit error rate (BER) performance by more than four orders of magnitude compared with the conventional simplified successive cancellation (SSC) decoding algorithm. Furthermore, in the mid-high SNR region, the average normalized complexity of the proposed algorithm is almost the same as that of the SSC decoding algorithm, while preserving the decoding performance gain.<\/jats:p>","DOI":"10.3390\/sym14061156","type":"journal-article","created":{"date-parts":[[2022,6,3]],"date-time":"2022-06-03T08:01:18Z","timestamp":1654243278000},"page":"1156","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Improving Decodability of Polar Codes by Adding Noise"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6452-5322","authenticated-orcid":false,"given":"Lingjun","family":"Kong","sequence":"first","affiliation":[{"name":"Faculty of Network and Telecommunication Engineering, Jinling Institute of Technology, Nanjing 211169, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Haiyang","family":"Liu","sequence":"additional","affiliation":[{"name":"Institute of Microelectronics of Chinese Academy of Sciences, Beijing 100029, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8834-9535","authenticated-orcid":false,"given":"Wentao","family":"Hou","sequence":"additional","affiliation":[{"name":"College of Telecommunications and Information Engineering, Nanjing University of Posts and Telecommunications, Nanjing 210003, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bin","family":"Dai","sequence":"additional","affiliation":[{"name":"School of Internet of Things, Nanjing University of Posts and Telecommunications, Nanjing 210003, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,6,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1117","DOI":"10.1109\/COMST.2022.3151028","article-title":"Cellular, wide-area, and non-terrestrial IoT: A survey on 5G advances and the road towards 6G","volume":"24","author":"Vaezi","year":"2022","journal-title":"IEEE Commun. Surv. Tutor."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Fang, Y., Bu, Y., Chen, P., Lau, F.C., and Al Otaibi, S. (2022). Irregular-mapped protograph LDPC-coded modulation: A bandwidth-efficient solution for 6G-enabled mobile networks. IEEE Trans. Intell. Transp. Syst., 1\u201314.","DOI":"10.1109\/TITS.2021.3122994"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"442","DOI":"10.1016\/j.comcom.2020.03.003","article-title":"Sensing and throughput analysis of a MU-MIMO based cognitive radio scheme for the internet of things","volume":"154","author":"Miah","year":"2020","journal-title":"Comput. Commun."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"75264","DOI":"10.1109\/ACCESS.2020.2988510","article-title":"Artificial intelligence in education: A review","volume":"8","author":"Chen","year":"2020","journal-title":"IEEE Access"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"60","DOI":"10.1109\/MVT.2018.2867640","article-title":"Channel coding in 5G new radio: A tutorial overview and performance comparison with 4G LTE","volume":"13","author":"Hui","year":"2018","journal-title":"IEEE Veh. Technol. Mag."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"270","DOI":"10.1109\/MNET.001.1900289","article-title":"Physical-layer network coding: An efficient technique for wireless communications","volume":"34","author":"Chen","year":"2020","journal-title":"IEEE Netw."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"11033","DOI":"10.1109\/TVT.2021.3106053","article-title":"Protograph LDPC-coded BICM-ID with irregular CSK mapping in visible light communication systems","volume":"70","author":"Dai","year":"2021","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1109\/COMST.2020.2967127","article-title":"Design of polar codes in 5G new radio","volume":"23","author":"Bioglio","year":"2021","journal-title":"IEEE Commun. Surv. Tutor."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Zhang, H., Li, R., Wang, J., Dai, S., Zhang, G., Chen, Y., Luo, H., and Wang, J. (2018, January 20\u201324). Parity-check polar coding for 5G and beyond. Proceedings of the IEEE International Conference on Communications (ICC), Kansas City, MO, USA.","DOI":"10.1109\/ICC.2018.8422462"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1166","DOI":"10.1109\/JPROC.2021.3061701","article-title":"6G wireless systems: Vision, requirements, challenges, insights, and opportunities","volume":"109","author":"Tataria","year":"2021","journal-title":"Proc. IEEE"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"3051","DOI":"10.1109\/TIT.2009.2021379","article-title":"Channel polarization: A method for constructing capacity-achieving codes for symmetric binary-input memoryless channels","volume":"55","author":"Arikan","year":"2009","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"6562","DOI":"10.1109\/TIT.2013.2272694","article-title":"How to construct polar codes","volume":"59","author":"Tal","year":"2012","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"3221","DOI":"10.1109\/TCOMM.2012.081512.110872","article-title":"Efficient design and decoding of polar codes","volume":"60","author":"Trifonov","year":"2012","journal-title":"IEEE Trans. Commun."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1378","DOI":"10.1109\/LCOMM.2011.101811.111480","article-title":"A simplified successive-cancellation decoder for polar codes","volume":"15","author":"Kschischang","year":"2011","journal-title":"IEEE Commun. Lett."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2213","DOI":"10.1109\/TIT.2015.2410251","article-title":"List decoding of polar codes","volume":"61","author":"Tal","year":"2015","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1668","DOI":"10.1109\/LCOMM.2012.090312.121501","article-title":"CRC-aided decoding of polar codes","volume":"16","author":"Niu","year":"2012","journal-title":"IEEE Commun. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"3100","DOI":"10.1109\/TCOMM.2013.070213.120789","article-title":"Improved successive cancellation decoding of polar codes","volume":"61","author":"Chen","year":"2013","journal-title":"IEEE Trans. Commun."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"6496","DOI":"10.1109\/TSP.2014.2366712","article-title":"Early stopping criteria for energy-efficient low-latency belief-propagation polar code decoders","volume":"62","author":"Yuan","year":"2014","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1109\/TSP.2012.2223693","article-title":"A semi-parallel successive-cancellation decoder for polar codes","volume":"61","author":"Leroux","year":"2013","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2044","DOI":"10.1109\/LCOMM.2012.111612.121898","article-title":"An adaptive successive cancellation list decoder for polar codes with cyclic redundancy check","volume":"16","author":"Li","year":"2012","journal-title":"IEEE Commun. Lett."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"5165","DOI":"10.1109\/TSP.2015.2439211","article-title":"LLR-based successive cancellation list decoding of polar codes","volume":"63","author":"Parizi","year":"2015","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Niu, K., Dai, J., Tan, K., and Gao, J. (2021, January 28\u201330). Deep learning methods for channel decoding: A brief tutorial. Proceedings of the IEEE\/CIC International Conference on Communications in China (ICCC), Xiamen, China.","DOI":"10.1109\/ICCC52777.2021.9580304"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Cammerer, S., Gruber, T., Hoydis, J., and Brink, S.T. (2017, January 4\u20138). Scaling deep learning-based decoding of polar codes via partitioning. Proceedings of the IEEE Global Communications Conference, Singapore.","DOI":"10.1109\/GLOCOM.2017.8254811"},{"key":"ref_24","unstructured":"Doan, N., Ali Hashemi, S., and Gross, W.J. (September, January 31). Neural successive cancellation decoding of polar codes. Proceedings of the IEEE International Workshop on Signal Processing Advances in Wireless Communications (SPAWC), London, UK."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Wang, X., Zhang, H., Li, R., Huang, L., Dai, S., Huangfu, Y., and Wang, J. (2019, January 8\u201311). Learning to flip successive cancellation decoding of polar codes with LSTM networks. Proceedings of the IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC), Istanbul, Turkey.","DOI":"10.1109\/PIMRC.2019.8904878"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1109\/JETCAS.2020.2995962","article-title":"Deep learning-aided belief propagation decoder for polar codes","volume":"10","author":"Xu","year":"2020","journal-title":"IEEE J. Emerg. Sel. Top. Circuits Syst."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"817","DOI":"10.1109\/TCCN.2019.2946358","article-title":"Exploiting error-correction-CRC for polar SCL decoding: A deep learning-based approach","volume":"6","author":"Liu","year":"2020","journal-title":"IEEE Trans. Cogn. Commun. Netw."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"8","DOI":"10.1109\/97.809511","article-title":"Can detectability be improved by adding noise?","volume":"7","author":"Kay","year":"2000","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1175\/1520-0469(1980)037<0245:SPOAPC>2.0.CO;2","article-title":"Stochastic perturbation of a pure connective motion","volume":"37","author":"Sutera","year":"2010","journal-title":"J. Atmos. Sci."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1285","DOI":"10.1109\/LCOMM.2019.2918535","article-title":"Noise-aided belief propagation list decoding of polar codes","volume":"23","author":"Arli","year":"2019","journal-title":"IEEE Commun. Lett."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Xiao, D., and Gu, Z. (2020, January 5\u201319). Dynamic perturbation decoding of polar-CRC cascaded code. Proceedings of the IEEE International Wireless Communications, and Mobile Computing, Limassol, Cyprus.","DOI":"10.1109\/IWCMC48107.2020.9148465"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1491","DOI":"10.1109\/LWC.2020.2994770","article-title":"Multi-round belief propagation decoding with impulsive perturbation for short LDPC codes","volume":"9","author":"Lee","year":"2020","journal-title":"IEEE Wirel. Commun. Lett."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Gerrar, N.K., Zhao, S., and Kong, L. (2018, January 18\u201320). A CRC-aided perturbed decoding of polar codes. Proceedings of the 14th International Conference on Wireless Communications, Networking and Mobile Computing (WiCOM 2018), Chongqing, China.","DOI":"10.12783\/dtcse\/wicom2018\/26296"},{"key":"ref_34","first-page":"1900","article-title":"A decoding algorithm of polar codes based on perturbation with CNN","volume":"43","author":"Zhao","year":"2021","journal-title":"J. Electron. Inf. Technol."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1859","DOI":"10.1049\/cmu2.12197","article-title":"Error correction in data storage systems using polar codes","volume":"15","author":"Gerrar","year":"2021","journal-title":"IET Commun."},{"key":"ref_36","unstructured":"Kong, L., Liu, H., Hou, W., and Meng, C. (Electron. Lett., 2022). A perturbed and directed neural-evolutionary decoding algorithm for polar codes, Electron. Lett., Submitted."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"L453","DOI":"10.1088\/0305-4470\/14\/11\/006","article-title":"The mechanism of stochastic resonance","volume":"14","author":"Benzi","year":"1981","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"McDonnell, M.D., and Abbott, D. (2009). What is stochastic resonance? definitions, misconceptions, debates, and its relevance to biology. PLoS Comput. Biol., 5.","DOI":"10.1371\/journal.pcbi.1000348"},{"key":"ref_39","unstructured":"Chen, H., Varshney, P.K., Michels, J.H., and Kay, S. (2006, January 14\u201319). Approaching near optimal detection performance via stochastic resonance. Proceedings of the IEEE International Conference on Acoustics Speech and Signal Processing Proceedings (ICASSP), Toulouse, France."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Shih, K., and Shiu, D. (2006, January 11\u201314). Perturbed decoding algorithm for concatenated error correcting and detecting codes system. Proceedings of the IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC), Helsinki, Finland.","DOI":"10.1109\/PIMRC.2006.254446"},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Wang, W., and Shiu, D. (2007, January 3\u20137). The theory behind perturbed decoding algorithm. Proceedings of the IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC), Athens, Greece.","DOI":"10.1109\/PIMRC.2007.4394497"},{"key":"ref_42","unstructured":"He, Z., Li, X., Wang, S., and Qi, Y. (2021, January 5\u20137). A noise-assisted polar code attempt decoding algorithm. Proceedings of the IEEE International Conference on Computer Research and Development (ICCRD), Beijing, China."},{"key":"ref_43","first-page":"1","article-title":"A study on iterative decoding with LLR modulator by neural network using adjacent track information in SMR system","volume":"55","author":"Nishikawa","year":"2019","journal-title":"IEEE Trans. Magn."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"5454","DOI":"10.1109\/TCSI.2020.3018048","article-title":"Flexible high throughput QC-LDPC decoder with perfect pipeline conflicts resolution and efficient hardware utilization","volume":"67","author":"Saranovac","year":"2020","journal-title":"IEEE Trans. Circuits Syst. I Regul. Pap."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"222572","DOI":"10.1109\/ACCESS.2020.3041290","article-title":"An improved SC flip decoding algorithm of polar codes based on genetic algorithm","volume":"8","author":"Wang","year":"2020","journal-title":"IEEE Access"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"4521","DOI":"10.1109\/TCOMM.2019.2908870","article-title":"Decoder-tailored polar code design using the genetic algorithm","volume":"67","author":"Elkelesh","year":"2019","journal-title":"IEEE Trans. Commun."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"3175","DOI":"10.1109\/LCOMM.2021.3104092","article-title":"Low-complexity construction of polar codes based on genetic algorithm","volume":"10","author":"Zhou","year":"2021","journal-title":"IEEE Commun. Lett."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"2227","DOI":"10.1109\/TCOMM.2022.3156074","article-title":"Belief propagation based joint detection and decoding for resistive random access memories","volume":"70","author":"Sun","year":"2022","journal-title":"IEEE Trans. Commun."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"26","DOI":"10.1109\/TCOMM.2019.2951403","article-title":"AI coding: Learning to construct error correction codes","volume":"68","author":"Huang","year":"2020","journal-title":"IEEE Trans. Commun."},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Moreloszaragoza, R.H. (2006). The Art of Error Correcting Coding, John Wiley & Sons. [2nd ed.].","DOI":"10.1002\/0470035706"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/6\/1156\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:24:10Z","timestamp":1760138650000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/6\/1156"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,6,3]]},"references-count":50,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2022,6]]}},"alternative-id":["sym14061156"],"URL":"https:\/\/doi.org\/10.3390\/sym14061156","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,6,3]]}}}