{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:30:16Z","timestamp":1760236216632,"version":"build-2065373602"},"reference-count":28,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2021,10,26]],"date-time":"2021-10-26T00:00:00Z","timestamp":1635206400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>This paper focuses on K-receiver discrete-time memoryless broadcast channels (DM-BCs) with private messages, where the transmitter wishes to convey K private messages to K receivers. A general inner bound on the capacity region is proposed based on an exhaustive message splitting and a K-level modified Marton\u2019s coding. The key idea is to split every message into \u2211j=1KKj\u22121 submessages each corresponding to a set of users who are assigned to recover them, and then send these submessages via codewords chosen from a K-level structure codebooks. To guarantee the joint typicality among all transmitted codewords, a sufficient condition on the subcodebooks\u2019 sizes is derived through a newly establishing hierarchical covering lemma, which extends the 2-level multivariate covering lemma to the K-level case with more intricate dependences. As the number of auxiliary random variables and rate conditions both increase exponentially with K, the standard Fourier\u2013Motzkin elimination procedure becomes infeasible when K is large. To tackle this problem, we obtain a closed form of achievable rate region with a special observation of disjoint unions of sets that constitute the power set of {1,\u22ef,K}. The proposed achievable rate region allows arbitrary input probability mass functions and improves over previously known achievable (closed form) rate regions for K-receiver (K\u22653) BCs.<\/jats:p>","DOI":"10.3390\/e23111408","type":"journal-article","created":{"date-parts":[[2021,10,26]],"date-time":"2021-10-26T23:49:52Z","timestamp":1635292192000},"page":"1408","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["On the Achievable Rate Region of the K-Receiver Broadcast Channels via Exhaustive Message Splitting"],"prefix":"10.3390","volume":"23","author":[{"given":"Rui","family":"Tang","sequence":"first","affiliation":[{"name":"School of Information Science and Technology, ShanghaiTech University, Shanghai 201210, China"},{"name":"Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences, Shanghai 200083, China"},{"name":"University of Chinese Academy of Sciences, Beijing 100049, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Songjie","family":"Xie","sequence":"additional","affiliation":[{"name":"School of Information Science and Technology, ShanghaiTech University, Shanghai 201210, China"},{"name":"Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences, Shanghai 200083, China"},{"name":"University of Chinese Academy of Sciences, Beijing 100049, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4383-9995","authenticated-orcid":false,"given":"Youlong","family":"Wu","sequence":"additional","affiliation":[{"name":"School of Information Science and Technology, ShanghaiTech University, Shanghai 201210, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,10,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2","DOI":"10.1109\/TIT.1972.1054727","article-title":"Broadcast Channels","volume":"18","author":"Cover","year":"1972","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"El Gamal, A., and Kim, Y.H. (2011). Network Information Theory, Cambridge University Press.","DOI":"10.1017\/CBO9781139030687"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2524","DOI":"10.1109\/18.720547","article-title":"Comments on Broadcast Channels","volume":"44","author":"Cover","year":"1998","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"306","DOI":"10.1109\/TIT.1979.1056046","article-title":"A Coding Theorem for the Discrete Memoryless Broadcast Channel","volume":"25","author":"Marton","year":"1979","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"60","DOI":"10.1109\/TIT.1977.1055655","article-title":"General Broadcast Channels with Degraded Message Sets","volume":"23","author":"Korner","year":"1977","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1361","DOI":"10.1109\/TIT.2018.2880241","article-title":"On the Evaluation of Marton\u2019s Inner Bound for Two-Receiver Broadcast Channels","volume":"65","author":"Anantharam","year":"2019","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Gohari, A., and Nair, C. (2020, January 21\u201326). New Outer Bounds for the Two-Receiver Broadcast Channel. Proceedings of the 2020 IEEE International Symposium on Information Theory (ISIT), Los Angeles, CA, USA.","DOI":"10.1109\/ISIT44484.2020.9174003"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"7719","DOI":"10.1109\/TIT.2018.2868934","article-title":"Common-Message Broadcast Channels With Feedback in the Nonasymptotic Regime: Full Feedback","volume":"64","author":"Trillingsgaard","year":"2018","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"7686","DOI":"10.1109\/TIT.2018.2868953","article-title":"Common-Message Broadcast Channels With Feedback in the Nonasymptotic Regime: Stop Feedback","volume":"64","author":"Trillingsgaard","year":"2018","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Trillingsgaard, K.F., Yang, W., Durisi, G., and Popovski, P. (2015, January 14\u201319). Broadcasting a Common Message with Variable-Length Stop-Feedback Codes. Proceedings of the 2015 IEEE International Symposium on Information Theory (ISIT), Hong Kong, China.","DOI":"10.1109\/ISIT.2015.7282907"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"3936","DOI":"10.1109\/TIT.2006.880064","article-title":"The Capacity Region of the Gaussian Multiple-Input Multiple-Output Broadcast Channel","volume":"52","author":"Weingarten","year":"2006","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_12","unstructured":"Tse, D.N. (July, January 29). Optimal Power Allocation over Parallel Gaussian Broadcast Channels. Proceedings of the IEEE International Symposium on Information Theory, Ulm, Germany."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1083","DOI":"10.1109\/18.915665","article-title":"Capacity and Optimal Resource Allocation for Fading Broadcast Channels. I. Ergodic Capacity","volume":"47","author":"Li","year":"2001","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1103","DOI":"10.1109\/18.915667","article-title":"Capacity and Optimal Resource Allocation for Fading Broadcast Channels. II. Outage Capacity","volume":"47","author":"Li","year":"2001","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_15","unstructured":"Viswanath, P., and Tse, D.N.C. (July, January 30). Sum Capacity of the Multiple Antenna Gaussian Broadcast Channel. Proceedings of the IEEE International Symposium on Information Theory, Lausanne, Switzerland."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1875","DOI":"10.1109\/TIT.2004.833336","article-title":"Sum Capacity of Gaussian Vector Broadcast Channels","volume":"50","author":"Yu","year":"2004","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"2658","DOI":"10.1109\/TIT.2003.817421","article-title":"Duality, Achievable Rates, and Sum-Rate Capacity of Gaussian MIMO Broadcast Channels","volume":"49","author":"Vishwanath","year":"2003","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1691","DOI":"10.1109\/TIT.2003.813523","article-title":"On the Achievable Throughput of a Multiantenna Gaussian Broadcast Channel","volume":"49","author":"Caire","year":"2003","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Salman, M., and Varanasi, M.K. (2019, January 7\u201312). Capacity Results via Message Merging and Superposition Coding in the K-Receiver Broadcast Channel with General Message Sets. Proceedings of the 2019 IEEE International Symposium on Information Theory (ISIT), Paris, France.","DOI":"10.1109\/ISIT.2019.8849564"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Romero, H., and Varanasi, M. (2016, January 10\u201315). Superposition Coding in the Combination Network. Proceedings of the 2016 IEEE International Symposium on Information Theory (ISIT), Barcelona, Spain.","DOI":"10.1109\/ISIT.2016.7541679"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Romero, H.P., and Varanasi, M.K. (2017, January 25\u201330). Rate Splitting and Superposition Coding for Concurrent Groupcasting over the Broadcast Channel: A General Framework. Proceedings of the 2017 IEEE International Symposium on Information Theory (ISIT), Aachen, Germany.","DOI":"10.1109\/ISIT.2017.8006857"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Ganguly, S., and Wang, L. (2021, January 11\u201315). Sliding-Window Gelfand\u2013Pinsker Coding: General K-User Broadcast Channels. Proceedings of the 2020 IEEE Information Theory Workshop (ITW), Riva del Garda, Italy.","DOI":"10.1109\/ITW46852.2021.9457616"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Nair, C., and Wang, Z.V. (July, January 28). On 3-Receiver Broadcast Channels with 2-Degraded Message Sets. Proceedings of the 2009 IEEE International Symposium on Information Theory, Seoul, Korea.","DOI":"10.1109\/ISIT.2009.5205420"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Nair, C., and El Gamal, A. (August, January 31). The Capacity Region of a Class of 3-Receiver Broadcast Channels with Degraded Message Sets. Proceedings of the 2008 IEEE International Symposium on Information Theory, St. Petersburg, Russia.","DOI":"10.1109\/ISIT.2008.4595279"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"4058","DOI":"10.1109\/TIT.2011.2145970","article-title":"The Capacity Region of the Three Receiver Less Noisy Broadcast Channel","volume":"57","author":"Nair","year":"2011","journal-title":"IEEE Trans. Inform. Theory"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Nair, C., and Xia, L. (2012, January 1\u20136). On Three-Receiver More Capable Channels. Proceedings of the 2012 IEEE International Symposium on Information Theory Proceedings, Cambridge, MA, USA.","DOI":"10.1109\/ISIT.2012.6284213"},{"key":"ref_27","unstructured":"Romero, H.P. (2014). Fundamental Limits of Network Communication with General Message Sets: A Combinatorial Approach. [Ph.D. Thesis, University of Colorado at Boulder]."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Davey, B.A., and Priestley, H.A. (2002). Introduction to Lattices and Order, Cambridge University Press.","DOI":"10.1017\/CBO9780511809088"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/11\/1408\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:24:21Z","timestamp":1760167461000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/11\/1408"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,10,26]]},"references-count":28,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2021,11]]}},"alternative-id":["e23111408"],"URL":"https:\/\/doi.org\/10.3390\/e23111408","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2021,10,26]]}}}