{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,22]],"date-time":"2025-10-22T03:01:18Z","timestamp":1761102078406,"version":"build-2065373602"},"reference-count":18,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2013,8,28]],"date-time":"2013-08-28T00:00:00Z","timestamp":1377648000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Future Internet"],"abstract":"<jats:p>In this paper, we address the implementation of physical layer network coding (PNC) based on compute and forward (CF) in relay networks. It is known that the maximum achievable rates in CF-based transmission is limited due to the channel approximations at the relay. In this work, we propose the integer forcing precoder (IFP), which bypasses this maximum rate achievability limitation. Our precoder requires channel state information (CSI) at the transmitter, but only that of the channel between the transmitter and the relay, which is a feasible assumption. The overall contributions of this paper are three-fold. Firstly, we propose an implementation of CF using IFP and prove that this implementation achieves higher rates as compared to traditional relaying schemes. Further, the probability of error from the proposed scheme is shown to have up to 2 dB of gain over the existent lattice network coding-based implementation of CF. Secondly, we analyze the two phases of transmission in the CF scheme, thereby characterizing the end-to-end behavior of the CF and not only one-phase behavior, as in previous proposals. Finally, we develop decoders for both the relay and the destination. We use a generalization of Bezout\u2019s theorem to justify the construction of these decoders. Further, we make an analytical derivation of the end-to-end probability of error for cubic lattices using the proposed scheme.<\/jats:p>","DOI":"10.3390\/fi5030439","type":"journal-article","created":{"date-parts":[[2013,8,28]],"date-time":"2013-08-28T12:53:31Z","timestamp":1377694411000},"page":"439-459","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Physical Layer Network Coding Based on Integer Forcing Precoded Compute and Forward"],"prefix":"10.3390","volume":"5","author":[{"given":"Smrati","family":"Gupta","sequence":"first","affiliation":[{"name":"Department of Telecommunications and Systems Engineering, Autonomous University of Barcelona, Bellaterra, Barcelona 08193, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"M.","family":"V\u00e1zquez-Castro","sequence":"additional","affiliation":[{"name":"Department of Telecommunications and Systems Engineering, Autonomous University of Barcelona, Bellaterra, Barcelona 08193, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2013,8,28]]},"reference":[{"key":"ref_1","unstructured":"Zhang, S., Liew, S., and Lam, P. Physical Layer Network Coding. arXiv.org Website. Available online: http:\/\/arxiv.org\/abs\/0704.2475."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"6463","DOI":"10.1109\/TIT.2011.2165816","article-title":"Compute and forward: Harnessing interference through structured codes","volume":"57","author":"Nazer","year":"2011","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Feng, C., Silva, D., and Kschischang, F. (2010, January 13\u201318). An Algebraic Approach to Physical-Layer Network Coding. Proceedings of 2010 IEEE International Symposium on Information Theory Proceedings (ISIT), Austin, TX, USA.","DOI":"10.1109\/ISIT.2010.5513739"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"5214","DOI":"10.1109\/TIT.2012.2197720","article-title":"The degrees of freedom of compute-and-forward","volume":"58","author":"Niesen","year":"2012","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Belfiore, J.-C. (2011, January 16\u201320). Lattice Codes for the Compute-and-Forward Protocol: The Flatness Factor. Proceedings of the IEEE Information Theory Workshop (ITW), Paraty, Brazil.","DOI":"10.1109\/ITW.2011.6089376"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"438","DOI":"10.1109\/JPROC.2010.2094170","article-title":"Reliable physical layer network coding","volume":"99","author":"Nazer","year":"2011","journal-title":"Proc. IEEE"},{"key":"ref_7","unstructured":"Chen, F., Silva, D., and Kschischang, F.R. (2012, January 1\u20136). Blind Compute-and-Forward. Proceedings of the IEEE International Symposium on Information Theory Proceedings (ISIT), Cambridge, MA, USA."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Hong, S.-N., and Caire, G. (2012, January 1\u20136). Reverse Compute and Forward: A Low-Complexity Architecture for Downlink Distributed Antenna Systems. Proceedings of the IEEE International Symposium on Information Theory Proceedings (ISIT), Cambridge, MA, USA.","DOI":"10.1109\/ISIT.2012.6283033"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"973","DOI":"10.1109\/TCOMM.2012.010213.120265","article-title":"A new physical-layer network coding scheme with eigen-direction alignment precoding for mimo two-way relaying","volume":"61","author":"Yang","year":"2013","journal-title":"IEEE Trans. Commun."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Gupta, S., and Vazquez-Castro, M.A. (2012, January 8\u201310). Physical-Layer Network Coding based on Integer-forcing Precoded Compute and Forward. Proceedings of the IEEE 8th International Conference on Wireless and Mobile Computing, Networking and Communications (WiMob), Barcelona, Spain.","DOI":"10.1109\/WiMOB.2012.6379137"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"4409","DOI":"10.1109\/TSP.2008.924638","article-title":"Zero-forcing precoding and generalized inverses","volume":"56","author":"Wiesel","year":"2008","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2806","DOI":"10.1109\/TSP.2005.850352","article-title":"On the sphere-decoding algorithm I. Expected complexity","volume":"53","author":"Hassibi","year":"2005","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Zamir, R. (2009, January 8\u201313). Lattices are Everywhere. Proceedings of the Information Theory and Applications Workshop, San Diego, CA, USA.","DOI":"10.1109\/ITA.2009.5044976"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1123","DOI":"10.1109\/18.21245","article-title":"Coset codes-I: Introduction and geometrical classification","volume":"34","author":"Forney","year":"1988","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Monteiro, F.A., and Wassell, I.J. (2011, January 14\u201317). Dual-Lattice-Aided MIMO Detection for Slow Fading Channel. Proceedings of the IEEE International Symposium on Signal Processing and Information Technology, Bilbao, Spain.","DOI":"10.1109\/ISSPIT.2011.6151613"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"670","DOI":"10.1109\/18.749010","article-title":"Universal bound on the performance of lattice codes","volume":"45","author":"Tarokh","year":"1999","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_17","first-page":"459","article-title":"Complexity of Bezout\u2019s theorem. I. Geometric aspects","volume":"6","author":"Shub","year":"1993","journal-title":"J. Am. Math. Soc."},{"key":"ref_18","unstructured":"Papoulis, A. (1984). Probability, Random Variables, and Stochastic Processes, McGraw-Hill. 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