{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,21]],"date-time":"2026-05-21T16:32:47Z","timestamp":1779381167047,"version":"3.53.1"},"reference-count":22,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2020,7,17]],"date-time":"2020-07-17T00:00:00Z","timestamp":1594944000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61901267"],"award-info":[{"award-number":["61901267"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The four-node relay broadcast channel (RBC) is considered, in which a transmitter communicates with two receivers with the assistance of a relay node. We first investigate three types of physically degraded RBCs (PDRBCs) based on different degradation orders among the relay and the receivers\u2019 observed signals. For the discrete memoryless (DM) case, only the capacity region of the second type of PDRBC is already known, while for the Gaussian case, only the capacity region of the first type of PDRBC is already known. In this paper, we step forward and make the following progress: (1) for the first type of DM-PDRBC, a new outer bound is established, which has the same rate expression as an existing inner bound, with only a slight difference on the input distributions; (2) for the second type of Gaussian PDRBC, the capacity region is established; (3) for the third type of PDRBC, the capacity regions are established both for DM and Gaussian cases. Besides, we also consider the RBC with relay feedback where the relay node can send the feedback signal to the transmitter. A new coding scheme based on a hybrid relay strategy and a layered Marton\u2019s coding is proposed. It is shown that our scheme can strictly enlarge Behboodi and Piantanida\u2019s rate region, which is tight for the second type of DM-PDRBC. Moreover, we show that capacity regions of the second and third types of PDRBCs are exactly the same as that without feedback, which means feedback cannot enlarge capacity regions for these types of RBCs.<\/jats:p>","DOI":"10.3390\/e22070784","type":"journal-article","created":{"date-parts":[[2020,7,17]],"date-time":"2020-07-17T10:22:02Z","timestamp":1594981322000},"page":"784","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["On the Capacity Regions of Degraded Relay Broadcast Channels with and without Feedback"],"prefix":"10.3390","volume":"22","author":[{"given":"Bingbing","family":"Hu","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 200050, China"},{"name":"University of Chinese Academy of Sciences, Beijing 100049, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ke","family":"Wang","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 200050, China"},{"name":"University of Chinese Academy of Sciences, Beijing 100049, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0854-4320","authenticated-orcid":false,"given":"Yingying","family":"Ma","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 200050, China"},{"name":"University of Chinese Academy of Sciences, Beijing 100049, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Youlong","family":"Wu","sequence":"additional","affiliation":[{"name":"School of Information Science and Technology, ShanghaiTech University, Shanghai 201210, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2020,7,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"120","DOI":"10.2307\/1426331","article-title":"Three-terminal communication channels","volume":"3","year":"1971","journal-title":"Adv. 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