{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:35:18Z","timestamp":1760243718388,"version":"build-2065373602"},"reference-count":20,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2012,9,28]],"date-time":"2012-09-28T00:00:00Z","timestamp":1348790400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>An achievable rate region for line networks with edge and node capacity constraints and broadcast channels (BCs) is derived. The region is shown to be the capacity region if the BCs are orthogonal, deterministic, physically degraded, or packet erasure with one-bit feedback. If the BCs are physically degraded with additive Gaussian noise then independent Gaussian inputs achieve capacity.<\/jats:p>","DOI":"10.3390\/e14101813","type":"journal-article","created":{"date-parts":[[2012,10,2]],"date-time":"2012-10-02T02:39:08Z","timestamp":1349145548000},"page":"1813-1828","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Network Coding for Line Networks with Broadcast Channels"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3904-9181","authenticated-orcid":false,"given":"Gerhard","family":"Kramer","sequence":"first","affiliation":[{"name":"Institute for Communications Engineering, Technische Universit\u00e4t M\u00fcnchen, 80333 Munich, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Seyed Mohammadsadegh Tabatabaei","family":"Yazdi","sequence":"additional","affiliation":[{"name":"Corporate R&D, Qualcomm, San Diego, CA 92121, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2012,9,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"4","DOI":"10.1109\/TIT.2002.806135","article-title":"Capacity results for the discrete memoryless network","volume":"49","author":"Kramer","year":"2003","journal-title":"IEEE Trans. Inform. Theor."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"4452","DOI":"10.1109\/TIT.2011.2146450","article-title":"Network coding in node-constrained line and star networks","volume":"57","author":"Savari","year":"2011","journal-title":"IEEE Trans. Inform. Theor."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1007\/s10922-005-9019-0","article-title":"Edge-cut bounds on network coding rates","volume":"14","author":"Kramer","year":"2006","journal-title":"J. Netw. Syst. Manag."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1109\/JSAC.2007.070213","article-title":"Spectral efficient protocols for half-duplex fading relay channels","volume":"25","author":"Rankov","year":"2007","journal-title":"IEEE J. Sel. Area. Comm."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"3517","DOI":"10.1109\/TIT.2007.904962","article-title":"Rate regions for relay broadcast channels","volume":"53","author":"Liang","year":"2007","journal-title":"IEEE Trans. Inform. Theor."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Kramer, G., and Shamai, S. (2007, January 2\u20136). Capacity for classes of broadcast channels with receiver side information. Proceedings of the IEEE Information Theory Workshop, Tahoe City, CA, USA.","DOI":"10.1109\/ITW.2007.4313093"},{"key":"ref_7","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. Inform. Theor."},{"key":"ref_8","unstructured":"Cover, T.M., and Thomas, J.A. (1991). Elements of Information Theory, John Wiley & Sons."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Bakshi, M., Effros, M., Gu, W., and Koetter, R. (2007, January 24\u201329). On network coding of independent and dependent sources in line networks. Proceedings of the IEEE International Symposium on Information Theory, Nice, France.","DOI":"10.1109\/ISIT.2007.4557370"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1561\/0100000028","article-title":"Topics in multi-user information theory","volume":"4","author":"Kramer","year":"2007","journal-title":"Found. Trends Commun. Inf. Theory"},{"key":"ref_11","unstructured":"Kramer, G., and Savari, S.A. (2006, January 6\u201310). Capacity bounds for relay networks. Presented at the Workshop on Information Theory and Applications, UCSD Campus, La Holla, CA, USA."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1109\/TIT.1978.1055885","article-title":"The feedback capacity of degraded broadcast channels","volume":"24","author":"Gamal","year":"1978","journal-title":"IEEE Trans. Inform. Theor."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"120","DOI":"10.1109\/TIT.1981.1056302","article-title":"A proof of Marton\u2019s coding theorem for the discrete memoryless broadcast channel","volume":"27","author":"Gamal","year":"1981","journal-title":"IEEE Trans. Inform. Theor."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Georgiadis, L., and Tassiulas, L. (2009, January 15\u201316). Broadcast erasure channel with feedback\u2014Capacity and algorithms. Proceedings of the Workshop on Network Coding, Theory, and Applications, Lausanne, Switzerland.","DOI":"10.1109\/NETCOD.2009.5191394"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"931","DOI":"10.1109\/TIT.2011.2173723","article-title":"On the capacity of 1-to-K broadcast packet erasure channels with channel output feedback","volume":"58","author":"Wang","year":"2012","journal-title":"IEEE Trans. Inform. Theor."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"623","DOI":"10.1109\/TIT.1984.1056935","article-title":"The capacity of the white Gaussian multiple access channel with feedback","volume":"30","author":"Ozarow","year":"1984","journal-title":"IEEE Trans. Inform. Theor."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1872","DOI":"10.1109\/TIT.2011.2110110","article-title":"Wireless network information flow: A deterministic approach","volume":"57","author":"Avestimehr","year":"2011","journal-title":"IEEE Trans. Inform. Theor."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"3132","DOI":"10.1109\/TIT.2011.2119930","article-title":"Noisy network coding","volume":"57","author":"Lim","year":"2011","journal-title":"IEEE Trans. Inform. Theor."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1561\/1300000004","article-title":"Cooperative communications","volume":"1","author":"Kramer","year":"2006","journal-title":"Found. Trends Netw."},{"key":"ref_20","unstructured":"Knopp, R. (, January February). Two-way radio networks with a star topology. Proceedings of the 2006 International Zurich Seminar on Communications, Zurich, Switzerland."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/14\/10\/1813\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:52:36Z","timestamp":1760219556000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/14\/10\/1813"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,9,28]]},"references-count":20,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2012,10]]}},"alternative-id":["e14101813"],"URL":"https:\/\/doi.org\/10.3390\/e14101813","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2012,9,28]]}}}