Non-binary GLD codes and their lattices

The recently discovered family of generalized low-density (GLD) lattices brings new mathematical challenges to coding theorists and practitioners. Given the excellent performance of integer GLD lattices in high dimensions and motivated by the simple lattice structure used for fast iterative decoding, this paper is a first attempt to analyze GLD lattices for asymptotically large dimensions. Firstly, we describe non-binary GLD codes and show their asymptotic goodness in terms of minimum Hamming distance. Secondly, we consider a GLD lattice ensemble built via Construction A from non-binary GLD codes, and analyze their goodness with respect to Poltyrev limit on the Gaussian channel. Finally, at large dimensions and using a large code alphabet, we prove that infinite GLD lattice constellations attain Poltyrev capacity limit under maximum likelihood decoding.

[1]  J. Boutros,et al.  A low complexity FEC scheme based on the intersection of interleaved block codes , 1999, 1999 IEEE 49th Vehicular Technology Conference (Cat. No.99CH36363).

[2]  F. MacWilliams,et al.  The Theory of Error-Correcting Codes , 1977 .

[3]  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.

[4]  G. David Forney,et al.  Coset codes-I: Introduction and geometrical classification , 1988, IEEE Trans. Inf. Theory.

[5]  J. Boutros,et al.  Generalized low density (Tanner) codes , 1999, 1999 IEEE International Conference on Communications (Cat. No. 99CH36311).

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

[7]  F. Lemmermeyer.  Error-correcting Codes , 2005 .

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

[9]  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.

[10]  Uri Erez,et al.  A simple proof for the existence of “good” pairs of nested lattices , 2012 .

[11]  Nicola di Pietro,et al.  On infinite and finite lattice constellations for the additive white Gaussian Noise Channel. (Constellations finies et infinies de réseaux de points pour le canal AWGN) , 2014 .

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

[13]  Joseph J. Boutros,et al.  Spectral Thinning in GLD Lattices , 2015 .

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

[15]  Joseph Jean Boutros,et al.  Generalized low-density (GLD) lattices , 2014, 2014 IEEE Information Theory Workshop (ITW 2014).

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

[17]  W. W. Peterson,et al.  Error-Correcting Codes. , 1962 .

[18]  Cong Ling,et al.  Achieving AWGN Channel Capacity With Lattice Gaussian Coding , 2014, IEEE Transactions on Information Theory.