Multistage compute-and-forward with multilevel lattice codes based on product constructions

Product construction with two levels proposed in [1] is a lattice construction which can be thought of as Construction A with codes that can be represented as the Cartesian product of two linear codes. This paper first generalizes the product construction to arbitrary number of levels. More importantly, the existence of a sequence of such lattices that are good for quantization and Poltyrev-good under multistage decoding is proved. This family of lattices is then used to generate a sequence of nested lattice codes based on the recent construction of Ordentlich and Erez. This allows one to achieve the same computation rate of Nazer and Gastpar for compute-and-forward with multistage decoding, which is termed multistage compute-and-forward.

[1]  R. Breusch,et al.  Zur Verallgemeinerung des Bertrandschen Postulates, daß zwischenx und 2x stets Primzahlen liegen , 1932 .

[2]  N. Sloane,et al.  Sphere Packings and Error-Correcting Codes , 1971, Canadian Journal of Mathematics.

[3]  Hideki Imai,et al.  A new multilevel coding method using error-correcting codes , 1977, IEEE Trans. Inf. Theory.

[4]  Gottfried Ungerboeck,et al.  Channel coding with multilevel/phase signals , 1982, IEEE Trans. Inf. Theory.

[5]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

[6]  Philippe Piret,et al.  Algebraic constructions of Shannon codes for regular channels , 1982, IEEE Trans. Inf. Theory.

[7]  N. Sloane,et al.  New Lattice Packings of Spheres , 1983, Canadian Journal of Mathematics.

[8]  W. Fischer,et al.  Sphere Packings, Lattices and Groups , 1990 .

[9]  Thomas M. Cover,et al.  Elements of Information Theory , 2005 .

[10]  N. J. A. Sloane,et al.  The On-Line Encyclopedia of Integer Sequences , 2003, Electron. J. Comb..

[11]  Shlomo Shamai,et al.  Performance analysis of a multilevel coded modulation system , 1994, IEEE Trans. Commun..

[12]  Gregory Poltyrev,et al.  On coding without restrictions for the AWGN channel , 1993, IEEE Trans. Inf. Theory.

[13]  Meir Feder,et al.  On lattice quantization noise , 1996, IEEE Trans. Inf. Theory.

[14]  Thomas C. Hales.  Sphere packings, I , 1997, Discret. Comput. Geom..

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

[16]  David J. C. MacKay,et al.  Low-density parity check codes over GF(q) , 1998, IEEE Communications Letters.

[17]  Robert F. H. Fischer,et al.  Multilevel codes: Theoretical concepts and practical design rules , 1999, IEEE Trans. Inf. Theory.

[18]  N. Sloane,et al.  Sphere Packing and Error-Correcting Codes , 1999 .

[19]  Sae-Young Chung,et al.  Sphere-bound-achieving coset codes and multilevel coset codes , 2000, IEEE Trans. Inf. Theory.

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

[21]  Michael Lentmaier,et al.  Convergence analysis for a class of LDPC convolutional codes on the erasure channel , 2004 .

[22]  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.

[23]  Uri Erez,et al.  Achieving 1/2 log (1+SNR) on the AWGN channel with lattice encoding and decoding , 2004, IEEE Transactions on Information Theory.

[24]  Michael Lentmaier,et al.  Terminated LDPC convolutional codes with thresholds close to capacity , 2005, Proceedings. International Symposium on Information Theory, 2005. ISIT 2005..

[25]  Simon Litsyn,et al.  Lattices which are good for (almost) everything , 2005, IEEE Transactions on Information Theory.

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

[27]  Rüdiger L. Urbanke,et al.  Modern Coding Theory , 2008 .

[28]  Sae-Young Chung,et al.  Capacity of the Gaussian Two-way Relay Channel to within 1/2 Bit , 2009, ArXiv.

[29]  Rüdiger L. Urbanke,et al.  Threshold Saturation via Spatial Coupling: Why Convolutional LDPC Ensembles Perform So Well over the BEC , 2010, IEEE Transactions on Information Theory.

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

[31]  Sae-Young Chung,et al.  Capacity of the Gaussian Two-Way Relay Channel to Within ${1\over 2}$ Bit , 2009, IEEE Transactions on Information Theory.

[32]  Michael Lentmaier,et al.  Iterative Decoding Threshold Analysis for LDPC Convolutional Codes , 2010, IEEE Transactions on Information Theory.

[33]  Frank R. Kschischang,et al.  An Algebraic Approach to Physical-Layer Network Coding , 2010, IEEE Transactions on Information Theory.

[34]  Urs Niesen,et al.  The Degrees of Freedom of Compute-and-Forward , 2011, IEEE Transactions on Information Theory.

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

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

[37]  Krishna R. Narayanan,et al.  Concatenated Signal Codes with Applications to Compute and Forward , 2011, 2011 IEEE Global Telecommunications Conference - GLOBECOM 2011.

[38]  Fernando Auil,et al.  An Algorithm to Generate Square-Free Numbers and to Compute the Moebius Function , 2011, ArXiv.

[39]  Nicolas Macris,et al.  A proof of threshold saturation for spatially-coupled LDPC codes on BMS channels , 2012, 2012 50th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[40]  Yu-Chih Huang,et al.  Lattices Over Eisenstein Integers for Compute-and-Forward , 2014, IEEE Transactions on Information Theory.

[41]  Loïc Brunel,et al.  Integer low-density lattices based on construction A , 2012, 2012 IEEE Information Theory Workshop.

[42]  Emanuele Viterbo,et al.  Construction of Barnes-Wall lattices from linear codes over rings , 2012, 2012 IEEE International Symposium on Information Theory Proceedings.

[43]  Uri Erez,et al.  A Simple Proof for the Existence of “Good” Pairs of Nested Lattices , 2012, IEEE Transactions on Information Theory.

[44]  Krishna R. Narayanan,et al.  Multilevel Coding Schemes for Compute-and-Forward With Flexible Decoding , 2013, IEEE Transactions on Information Theory.

[45]  Joseph Jean Boutros,et al.  New results on Construction A lattices based on very sparse parity-check matrices , 2013, 2013 IEEE International Symposium on Information Theory.

[46]  Yanfei Yan,et al.  Polar lattices: Where Arıkan meets Forney , 2013, 2013 IEEE International Symposium on Information Theory.

[47]  Henry D. Pfister,et al.  Spatially-coupled low density lattices based on construction a with applications to compute-and-forward , 2013, 2013 IEEE Information Theory Workshop (ITW).

[48]  Yu-Chih Huang,et al.  Lattice codes based on product constructions over F2q with applications to compute-and-forward , 2013, 2013 IEEE Information Theory Workshop (ITW).

[49]  Wen Chen,et al.  Integer-Forcing Linear Receiver Design with Slowest Descent Method , 2013, IEEE Transactions on Wireless Communications.

[50]  Yanfei Yan,et al.  Polar Lattices: Where Arikan Meets Forney , 2013, ArXiv.

[51]  Uri Erez,et al.  On the Robustness of Lattice Interference Alignment , 2013, IEEE Transactions on Information Theory.

[52]  Rüdiger L. Urbanke,et al.  Spatially coupled ensembles universally achieve capacity under belief propagation , 2012, 2012 IEEE International Symposium on Information Theory Proceedings.

[53]  Frédérique E. Oggier,et al.  Connections between Construction D and related constructions of lattices , 2013, Des. Codes Cryptogr..

[54]  Uri Erez,et al.  The Approximate Sum Capacity of the Symmetric Gaussian $K$ -User Interference Channel , 2012, IEEE Transactions on Information Theory.

[55]  Uri Erez,et al.  The Approximate Sum Capacity of the Symmetric Gaussian $K$ -User Interference Channel , 2014, IEEE Trans. Inf. Theory.

[56]  Michael Gastpar,et al.  Integer-forcing linear receivers , 2010, 2010 IEEE International Symposium on Information Theory.

[57]  Giuseppe Caire,et al.  Structured Lattice Codes for Some Two-User Gaussian Networks With Cognition, Coordination, and Two Hops , 2013, IEEE Transactions on Information Theory.