# American Institute of Mathematical Sciences

November  2010, 4(4): 441-452. doi: 10.3934/amc.2010.4.441

## On cycle-free lattices with high rate label codes

 1 Department of Mathematics and Computer Science, Amirkabir University of Technology, No. 424, Hafez Avenue, Tehran 15914, Iran, Iran

Received  October 2009 Revised  June 2010 Published  November 2010

Etzion et al. have shown that high rate codes based on cycle-free Tanner graphs have minimum distance at most $2$. This result was extended by Sadeghi et al. to a small class of lattices based on Construction $D'$ only. In this paper, we prove a key theorem which relates the minimum distance of every lattice to the minimum distance of its label code. Then, using this powerful tool along with some new bounds on minimum distance of cycle-free group codes, we generalize those results to a large class of lattices here called RPS and PFP lattices. More importantly, we show that this class of cycle-free lattices are not so good in the view of coding gain.
Citation: Amin Sakzad, Mohammad-Reza Sadeghi. On cycle-free lattices with high rate label codes. Advances in Mathematics of Communications, 2010, 4 (4) : 441-452. doi: 10.3934/amc.2010.4.441
##### References:
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##### References:
 [1] A. H. Banihashemi and F. R. Kschischang, Tanner graphs for block codes and lattices: construction and complexity, IEEE Trans. Information Theory, 47 (2001), 822-834. doi: 10.1109/18.910592.  Google Scholar [2] R. de Buda, Some optimal codes have structre, IEEE Trans. Information Theory, 7 (1989), 893-899. Google Scholar [3] Y.-S. Choi, I.-J. Baik and S.-Y. Chung, Iterative decoding for low-density parity-check lattices, ICACT, (2008), 358-361. Google Scholar [4] J. H. Conway and N. J. A. Sloane, "Sphere Packing, Lattices and Groups," 3rd edition, Springer-Verlag, New York, 1998.  Google Scholar [5] T. Etzion, A. Trachtenberg and A. Vardy, Which codes have cycle-free Tanner graphs?, IEEE Trans. Information Theory, 45 (1999), 2173-2181. doi: 10.1109/18.782170.  Google Scholar [6] G. D. Forney, Jr., Density/length profiles and trellis complexity of lattices, IEEE Trans. Information Theory, 40 (1994), 1753-1774. doi: 10.1109/18.340453.  Google Scholar [7] G. D. Forney, Jr., M. D. Trott and S. Chung, Sphere-bound-achieving coset codes and multilevel coset codes, IEEE Trans. Information Theory, 46 (2000), 820-850. doi: 10.1109/18.841165.  Google Scholar [8] R. G. Gallager, Low-density parity-check codes, Ire Trans. Information Theory, IT-8 (1962), 21-28. doi: 10.1109/TIT.1962.1057683.  Google Scholar [9] X.-Y. Hu, E. Eleftherou and D. M. Arnold, Regular and irregular progressive edge-growth Tanner graphs, IEEE Trans. Information Theory, 51 (2005), 386-398. doi: 10.1109/TIT.2004.839541.  Google Scholar [10] F. R. Kschischang, B. J. Frey and H. Loeliger, Factor graphs and the sum-product algorithm, IEEE Trans. Information Theory, 47 (2001), 498-519. doi: 10.1109/18.910572.  Google Scholar [11] M. R. Sadeghi, A. H. Banihashemi and D. Panario, Low-density parity-check lattices: construction and decoding complexity, IEEE Trans. Information Theory, 52 (2006), 4481-4495. doi: 10.1109/TIT.2006.881720.  Google Scholar [12] M. R. Sadeghi and D. Panario, Low density parity check lattices based on construction $D'$ and cycle-free Tanner graphs, Algebraic Coding Theory and Information Theory, AMS DIMACS, 28 (2005), 85-95.  Google Scholar [13] N. Sommer, M. Feder and O. Shalvi, Low-density lattice codes, IEEE Trans. Information Theory, 54 (2008), 1561-1586. doi: 10.1109/TIT.2008.917684.  Google Scholar [14] R. M. Tanner, A recursive approach to low-complexity codes, IEEE Trans. Information Theory, 27 (1981), 533-547. doi: 10.1109/TIT.1981.1056404.  Google Scholar [15] N. Wiberg, "Codes and Decoding on General Graphs," Ph.D thesis, Univ. Linköping, Linköping, Sweden, 1996. Google Scholar
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