{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,12]],"date-time":"2026-05-12T16:38:29Z","timestamp":1778603909076,"version":"3.51.4"},"reference-count":26,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2025,5,26]],"date-time":"2025-05-26T00:00:00Z","timestamp":1748217600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100002858","name":"China Postdoctoral Science Foundation","doi-asserted-by":"publisher","award":["2024M764138"],"award-info":[{"award-number":["2024M764138"]}],"id":[{"id":"10.13039\/501100002858","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002858","name":"China Postdoctoral Science Foundation","doi-asserted-by":"publisher","award":["62401048"],"award-info":[{"award-number":["62401048"]}],"id":[{"id":"10.13039\/501100002858","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002858","name":"China Postdoctoral Science Foundation","doi-asserted-by":"publisher","award":["62271044"],"award-info":[{"award-number":["62271044"]}],"id":[{"id":"10.13039\/501100002858","id-type":"DOI","asserted-by":"publisher"}]},{"name":"National Natural Science Foundation of China","award":["2024M764138"],"award-info":[{"award-number":["2024M764138"]}]},{"name":"National Natural Science Foundation of China","award":["62401048"],"award-info":[{"award-number":["62401048"]}]},{"name":"National Natural Science Foundation of China","award":["62271044"],"award-info":[{"award-number":["62271044"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Compared with scalar linear network coding (LNC) formulated over the finite field GF(2L), vector LNC offers enhanced flexibility in the code design by enabling linear operations over the vector space GF(2)L and demonstrates a number of advantages over scalar LNC. While random LNC (RLNC) has shown significant potential to improve the completion delay performance in wireless broadcasts, most prior studies focus on scalar RLNC. In particular, it is well known that, with increasing L, primitive scalar RLNC over GF(2L) asymptotically achieves the optimal completion delay. However, the completion delay performance of primitive vector RLNC remains unexplored. This work aims to fill in this blank. We derive closed-form expressions for the probability distribution and the expected value of both the completion delay at a single receiver and the system completion delay. We further unveil a fundamental limitation that is different from scalar RLNC: even for large enough L, primitive vector RLNC over GF(2)L inherently fails to reach optimal completion delay. In spite of this, the gap between the expected completion delay at a receiver and the optimal one is shown to be a constant smaller than 0.714, which implies that the expected completion delay normalized by the number P of original packets is asymptotically optimal with increasing P. We also validate our theoretical characterization through numerical simulations. Our theoretical characterization establishes primitive vector RLNC as a performance baseline for the future design of practical vector RLNC schemes with different design goals.<\/jats:p>","DOI":"10.3390\/e27060559","type":"journal-article","created":{"date-parts":[[2025,5,28]],"date-time":"2025-05-28T08:12:47Z","timestamp":1748419967000},"page":"559","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["On Vector Random Linear Network Coding in Wireless Broadcasts"],"prefix":"10.3390","volume":"27","author":[{"given":"Rina","family":"Su","sequence":"first","affiliation":[{"name":"School of Cyberspace Science and Technology, Beijing Institute of Technology, Beijing 100081, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Chengji","family":"Zhao","sequence":"additional","affiliation":[{"name":"School of Computer and Communication Engineering, University of Science and Technology Beijing, Beijing 100083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3213-1569","authenticated-orcid":false,"given":"Qifu","family":"Sun","sequence":"additional","affiliation":[{"name":"School of Computer and Communication Engineering, University of Science and Technology Beijing, Beijing 100083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhongshan","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Cyberspace Science and Technology, Beijing Institute of Technology (ZhuHai), Zhuhai 519088, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,5,26]]},"reference":[{"key":"ref_1","unstructured":"Medard, M., Effros, M., Karger, D., and Ho, T. (2003, January 1\u20133). On coding for non-multicast networks. Proceedings of the 41st Allerton Annual Conference on Communication, Control and Computing, Monticello, IL, USA."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"996","DOI":"10.1109\/TIT.2010.2094930","article-title":"Algebraic algorithms for vector network coding","volume":"57","author":"Ebrahimi","year":"2011","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"5096","DOI":"10.1109\/TCOMM.2016.2613085","article-title":"On vector linear solvability of multicast networks","volume":"64","author":"Sun","year":"2016","journal-title":"IEEE Trans. Commun."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2460","DOI":"10.1109\/TIT.2018.2797183","article-title":"Vector network coding based on subspace codes outperforms scalar linear network coding","volume":"64","author":"Etzion","year":"2018","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"6786","DOI":"10.1109\/TIT.2020.2995845","article-title":"Network-coding solutions for minimal combination networks and their sub-networks","volume":"66","author":"Cai","year":"2020","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"5580","DOI":"10.1109\/TIT.2021.3065364","article-title":"On the gap between scalar and vector solutions of generalized combination networks","volume":"67","author":"Liu","year":"2021","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1109\/TIT.2018.2832624","article-title":"Circular-shift linear network coding","volume":"65","author":"Tang","year":"2019","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"5606","DOI":"10.1109\/TCOMM.2020.3001133","article-title":"Delay-complexity trade-off of random linear network coding in wireless broadcast","volume":"68","author":"Su","year":"2020","journal-title":"IEEE Trans. Commun."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"4413","DOI":"10.1109\/TIT.2006.881746","article-title":"A random linear network coding approach to multicast","volume":"52","author":"Ho","year":"2006","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2547","DOI":"10.1109\/TCOMM.2018.2801791","article-title":"Reliability of multicast under random linear network coding","volume":"66","author":"Tsimbalo","year":"2018","journal-title":"IEEE Trans. Commun."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"7050","DOI":"10.1109\/TVT.2017.2670178","article-title":"Decoding delay performance of random linear network coding for broadcast","volume":"66","author":"Chatzigeorgiou","year":"2017","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"6328","DOI":"10.1109\/TIT.2013.2271895","article-title":"Throughput-delay analysis of random linear network coding for wireless broadcasting","volume":"59","author":"Swapna","year":"2013","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1109\/TCOMM.2015.2503398","article-title":"Analysis and optimization of sparse random linear network coding for reliable multicast services","volume":"64","author":"Tassi","year":"2016","journal-title":"IEEE Trans. Commun."},{"key":"ref_14","first-page":"766","article-title":"Adaptive causal network coding with feedback for multipath multi-hop communications","volume":"69","author":"Cohen","year":"2020","journal-title":"IEEE Trans. Commun."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"122","DOI":"10.1109\/MNET.2018.1700378","article-title":"Efficient coastal communications with sparse network coding","volume":"32","author":"Li","year":"2018","journal-title":"IEEE Netw."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"7828","DOI":"10.1109\/JIOT.2019.2908415","article-title":"Cooperative NOMA broadcasting\/multicasting for low-latency and high-reliability 5G cellular V2X communications","volume":"6","author":"Liu","year":"2019","journal-title":"IEEE Internet Things J."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Tang, H., Liu, H., Jin, S., Liu, W., and Sun, Q. (2024). On matrix representation of extension field GF(2L) an its application in vector linear network coding. Entropy, 26.","DOI":"10.3390\/e26100822"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2276","DOI":"10.1109\/TWC.2018.2791500","article-title":"Blind instantly decodable network codes for wireless broadcast of real-time multimedia","volume":"17","author":"Arefi","year":"2018","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_19","first-page":"77","article-title":"On the packet decoding delay of linear network coded wireless broadcast","volume":"46","author":"Yu","year":"2023","journal-title":"Can. J. Elect. Comput. Eng."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1035","DOI":"10.1109\/TPDS.2023.3237989","article-title":"Minimizing the average packet access time of the application layer for buffered instantly decodable network coding","volume":"34","author":"Mei","year":"2023","journal-title":"IEEE Trans. Parallel Distrib. Syst."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"16208","DOI":"10.1109\/TVT.2023.3298006","article-title":"Buffering instantly decodable network coding for a heterogenous cellular network with dual interfaces","volume":"72","author":"Mei","year":"2023","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"7843","DOI":"10.1109\/TCOMM.2022.3220867","article-title":"Completion delay of random linear network coding in full-duplex relay networks","volume":"70","author":"Su","year":"2022","journal-title":"IEEE Trans. Commun."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"3943","DOI":"10.1109\/TCOMM.2014.2364198","article-title":"P. From instantly decodable to random linear network coded broadcast","volume":"40","author":"Yu","year":"2014","journal-title":"IEEE Trans. Commun."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Giacaglia, G., Shi, X., Kim, M., Lucani, D.E., and Medard, M. (2013, January 9\u201313). Systematic network coding with the aid of a full-duplex relay. Proceedings of the 2013 IEEE International Conference on Communications (ICC), Budapest, Hungary.","DOI":"10.1109\/ICC.2013.6655057"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1186\/s13634-015-0273-3","article-title":"Systematic network coding for twohop lossy transmissions","volume":"2015","author":"Li","year":"2015","journal-title":"EURASIP J. Adv. Signal Process."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2330","DOI":"10.1109\/TIT.2011.2177562","article-title":"On coding for delay\u2014Network coding for time-division duplexing","volume":"58","author":"Lucani","year":"2012","journal-title":"IEEE Trans. Inf. Theory"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/6\/559\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:40:24Z","timestamp":1760031624000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/6\/559"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,5,26]]},"references-count":26,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2025,6]]}},"alternative-id":["e27060559"],"URL":"https:\/\/doi.org\/10.3390\/e27060559","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,5,26]]}}}