An algebraic approach to physical-layer network coding

The problem of designing new physical-layer network coding (PNC) schemes via lattice partitions is considered. Building on a recent work by Nazer and Gastpar, who demonstrated its asymptotic gain using information-theoretic tools, we take an algebraic approach to show its potential in non-asymptotic settings. We first relate Nazer-Gastpar's approach to the fundamental theorem of finitely generated modules over a principle ideal domain. Based on this connection, we generalize their code construction and simplify their encoding and decoding methods. This not only provides a transparent understanding of their approach, but more importantly, it opens up the opportunity to design efficient and practical PNC schemes. Finally, we apply our framework for PNC to a Gaussian relay network and demonstrate its advantage over conventional PNC schemes.

[1]  Suhas N. Diggavi,et al.  Approximately achieving Gaussian relay network capacity with lattice codes , 2010, 2010 IEEE International Symposium on Information Theory.

[2]  N. J. A. Sloane,et al.  A fast encoding method for lattice codes and quantizers , 1983, IEEE Trans. Inf. Theory.

[3]  Michael Gastpar,et al.  Compute-and-Forward: Harnessing Interference Through Structured Codes , 2009, IEEE Transactions on Information Theory.

[4]  Toshiaki Koike-Akino,et al.  Optimized constellations for two-way wireless relaying with physical network coding , 2009, IEEE Journal on Selected Areas in Communications.

[5]  Meir Feder,et al.  Shaping methods for low-density lattice codes , 2009, 2009 IEEE Information Theory Workshop.

[6]  Uri Erez,et al.  Achieving the gains promised by Integer-Forcing equalization with binary codes , 2010, 2010 IEEE 26-th Convention of Electrical and Electronics Engineers in Israel.

[7]  Justin Dauwels,et al.  Power-constrained communications using LDLC lattices , 2009, 2009 IEEE International Symposium on Information Theory.

[8]  Jean-Claude Belfiore Lattice codes for the compute-and-forward protocol: The flatness factor , 2011, 2011 IEEE Information Theory Workshop.

[9]  Alexander Sprintson,et al.  Joint Physical Layer Coding and Network Coding for Bidirectional Relaying , 2008, IEEE Transactions on Information Theory.

[10]  Petar Popovski,et al.  The Anti-Packets Can Increase the Achievable Throughput of a Wireless Multi-Hop Network , 2006, 2006 IEEE International Conference on Communications.

[11]  Frank R. Kschischang,et al.  Lattice network coding via signal codes , 2011, 2011 IEEE International Symposium on Information Theory Proceedings.

[12]  Frank R. Kschischang,et al.  Coding for Errors and Erasures in Random Network Coding , 2008, IEEE Trans. Inf. Theory.

[13]  Daniel Panario,et al.  Turbo Lattices: Construction and Performance Analysis , 2011, ArXiv.

[14]  Michael Gastpar,et al.  Practical code design for compute-and-forward , 2011, 2011 IEEE International Symposium on Information Theory Proceedings.

[15]  Jean-Claude Belfiore,et al.  Managing interference through space-time codes, lattice reduction and network coding , 2010, 2010 IEEE Information Theory Workshop on Information Theory (ITW 2010, Cairo).

[16]  William C. Brown,et al.  Matrices over commutative rings , 1993 .

[17]  Petar Popovski,et al.  Physical Network Coding in Two-Way Wireless Relay Channels , 2007, 2007 IEEE International Conference on Communications.

[18]  Sae-Young Chung,et al.  Capacity Bounds for Two-Way Relay Channels , 2008, 2008 IEEE International Zurich Seminar on Communications.

[19]  Daniel J. Costello,et al.  Channel coding: The road to channel capacity , 2006, Proceedings of the IEEE.

[20]  Michael Gastpar,et al.  Reliable Physical Layer Network Coding , 2011, Proceedings of the IEEE.

[21]  Michael Gastpar,et al.  Computing over Multiple-Access Channels with Connections to Wireless Network Coding , 2006, 2006 IEEE International Symposium on Information Theory.

[22]  Hans-Andrea Loeliger Averaging bounds for lattices and linear codes , 1997, IEEE Trans. Inf. Theory.

[23]  Johannes Blömer,et al.  Closest Vectors, Successive Minima, and Dual HKZ-Bases of Lattices , 2000, ICALP.

[24]  N. J. A. Sloane,et al.  Sphere Packings, Lattices and Groups , 1987, Grundlehren der mathematischen Wissenschaften.

[25]  Muriel Médard,et al.  An algebraic approach to network coding , 2003, TNET.

[26]  Sae-Young Chung,et al.  Noisy Network Coding , 2010, IEEE Transactions on Information Theory.

[27]  Meir Feder,et al.  Signal Codes: Convolutional Lattice Codes , 2011, IEEE Transactions on Information Theory.

[28]  G. David Forney,et al.  On the role of MMSE estimation in approaching the information-theoretic limits of linear Gaussian channels: Shannon meets Wiener , 2004, ArXiv.

[29]  Randall Dougherty,et al.  Insufficiency of linear coding in network information flow , 2005, IEEE Transactions on Information Theory.

[30]  Wai Ho Mow,et al.  Complex Lattice Reduction Algorithm for Low-Complexity Full-Diversity MIMO Detection , 2009, IEEE Transactions on Signal Processing.

[31]  G. David Forney,et al.  Convolutional codes I: Algebraic structure , 1970, IEEE Trans. Inf. Theory.

[32]  Michael Gastpar,et al.  MIMO compute-and-forward , 2009, 2009 IEEE International Symposium on Information Theory.

[33]  Suhas N. Diggavi,et al.  Wireless Network Information Flow: A Deterministic Approach , 2009, IEEE Transactions on Information Theory.

[34]  Krishna Narayanan,et al.  Power allocation strategies and lattice based coding schemes for bi-directional relaying , 2009, 2009 IEEE International Symposium on Information Theory.

[35]  Bernard R. McDonald,et al.  Linear Algebra over Commutative Rings , 1984 .

[36]  Shuo-Yen Robert Li,et al.  Linear network coding , 2003, IEEE Trans. Inf. Theory.

[37]  C. A. Rogers,et al.  An Introduction to the Geometry of Numbers , 1959 .

[38]  Krishna R. Narayanan,et al.  Multilevel coding schemes for compute-and-forward , 2011, 2011 IEEE International Symposium on Information Theory Proceedings.

[39]  Meir Feder,et al.  Low Density Lattice Codes , 2006, ISIT.

[40]  Gregory W. Wornell,et al.  Lattice-reduction-aided detectors for MIMO communication systems , 2002, Global Telecommunications Conference, 2002. GLOBECOM '02. IEEE.

[41]  Meir Feder,et al.  Signal codes , 2008, Proceedings 2003 IEEE Information Theory Workshop (Cat. No.03EX674).

[42]  Toshiaki Koike-Akino,et al.  Coded Bidirectional Relaying in Wireless Networks , 2009 .

[43]  Aydin Sezgin,et al.  Information Theory Capacity of the two-way relay channel within a constant gap , 2010, Eur. Trans. Telecommun..

[44]  Soung Chang Liew,et al.  Hot topic: physical-layer network coding , 2006, MobiCom '06.

[45]  Soung Chang Liew,et al.  > Replace This Line with Your Paper Identification Number (double-click Here to Edit) < 1 , 2022 .

[46]  G. David Forney,et al.  Multidimensional constellations. II. Voronoi constellations , 1989, IEEE J. Sel. Areas Commun..