Construction of lattices from LDPC codes

Low-density parity-check (LDPC) codes can have a good performance under iterative decoding algorithms. This idea is used to construct a class of lattices with relatively high coding gain and low decoding complexity. To construct such lattices, Construction D' is applied to the set of parity check vectors of a class of nested LDPC codes. Bounds on the minimum distance and the coding gain of the corresponding lattice are provided. We also provide a practical way of finding the cross sections of the lattice given the parity check matrix of a lattice. The progressive edge growth algorithm is extended and is used to construct a class of nested binary codes to generate the corresponding lattice. Simulation results confirm the good performance of this class of lattice.

[1]  Robert Michael Tanner,et al.  A recursive approach to low complexity codes , 1981, IEEE Trans. Inf. Theory.

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

[3]  Rudi de Buda,et al.  Some optimal codes have structure , 1989, IEEE J. Sel. Areas Commun..

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

[5]  G. David Forney.  Density/length profiles and trellis complexity of lattices codes , 1994, IEEE Trans. Inf. Theory.

[6]  G. David Forney,et al.  Dimension/length profiles and trellis complexity of linear block codes , 1994, IEEE Trans. Inf. Theory.

[7]  David J. C. MacKay,et al.  Good Error-Correcting Codes Based on Very Sparse Matrices , 1997, IEEE Trans. Inf. Theory.

[8]  Alexander Vardy,et al.  Universal Bound on the Performance of Lattice Codes , 1999, IEEE Trans. Inf. Theory.

[9]  Frank R. Kschischang,et al.  Tanner graphs for group block codes and lattices: Construction and complexity , 2001, IEEE Trans. Inf. Theory.

[10]  Evangelos Eleftheriou,et al.  Progressive edge-growth Tanner graphs , 2001, GLOBECOM'01. IEEE Global Telecommunications Conference (Cat. No.01CH37270).

[11]  M.R. Sadeghi,et al.  Iterative decoding algorithm of lattices , 2004, Canadian Conference on Electrical and Computer Engineering 2004 (IEEE Cat. No.04CH37513).

[12]  Mohammad-Reza Rafsanjani-Sadeghi.  Low density parity check lattices , 2004 .