Abstract
Low-Density Parity-Check codes (LDPCC) with Iterative Belief Propagation (Message Passing) decoding are attractive alternatives to Turbo codes. LDPCC previously discussed in the literature have involved matrices constructed using random techniques. In this paper, we discuss construction techniques for LDPCC involving multiple permutation matrices, each specified by a linear congruence. Construction options depend on the size of the parity-check matrix and the rate of the code. We relate desirable properties of the code to the parameters defining the linear congruences specifying the permutation matrices used to construct the code. For example, codes with few or no 4-cycles can be readily constructed. We summarize the construction options and describe selection processes for the parameters of the congruences. We then provide performance results for regular parity-check matrices constructed by random and the linear-congruence techniques for rate 1/2 transmit block-size 980 and rate 4/7 transmit block-size 847 codes. We introduce a symmetric channel model for decoding with the iterative belief propagation algorithm and describe its use as a heuristic for deciding whether a code is likely better or worse than most codes of the given rate and block size.
The authors would like to thank the reviewer and Pascal O. Vontobel for useful comments.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
J.W. Bond, S. Hui, H. Schmidt, Low Density Parity Check Codes based on Sparse Matrices with No Small Cycles, Institute of Mathematics Applications, Coding and Cryptography Conference, December 1997.
J.W. Bond, S. Hui, H. Schmidt, Decoding Low-Density Generator Matrix Codes with the Aid of Comma-free Source Codes, Proceedings of 1988 Information Theory Workshop, Killarney, Ireland, 22–26 June 1998.
J.W. BOND, S. Hui, H. Schmidt, Constructing low-density parity-check codes with circulant matrices, 1999 IEEE International Symposium on Information Theory, Metsovo, Greece, June 27-July 1, 1999.
J.W. Bond, S. Hui, H. Schmidt, The Euclidean Algorithm and Primitive Polynomials over Finite Fields, to be published in a book associated with AAECC-13 on Applied Algebra, Fall 1999.
J.W. Bond, S. Hui, H. Schmidt, Belief Propagation Decoding for Gaussian Channels, September 21, 1998.
J.F. Cheng and R.J. McEliece, Some High-Rate Near Capacity Codes for the Gaussian Channel, presented at the 34th Allerton Conference on Communication, Control, and Computing, 1996.
M.C. Davey and D.J.C. MacKay, Monte Carlo Simulations of Infinite Low Density Parity-Check Codes over GF(q), March 30, 1998, Proceedings of 1988 Information Theory Workshop, Killarney, Ireland, 22-26 June 1998.
R. Gallager, Low-Density Parity-Check Codes, MIT Press, Cambridge Mass, July 1963.
G.H. Hardy, and E.M. Wright, An Introduction to Number Theory, Fourth Edition, Clarendon Press, Oxford, 1968.
L.K. Hua, Introduction to Number Theory, Springer-Verlag, Berlin, 1982.
D.J.C. MacKay, Good Error-Correcting Codes Based on Very Sparse Matrices, IEEE Transactions on Information Theory, pp 399–431, Vol. 45, No. 2, March 1999.
D.J.C. MacKay and M.C. Davey, Evaluation of Gallager Codes for Short Block Length and High Rate Applications, Preprint.
T. Richardson, A. Shokrollahi and R. Urbanke, Design of Provably Good Low-Density Parity-Check Codes, 1999, Preprint.
T. Richardson and R. Urbanke, The Capacity of Low-Density Parity Check Codes under Message-Passing Decoding, Preprint.
B.M. Stewart, Theory of Numbers, Second Edition, Macmillan, New York, 1964.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag New York, Inc.
About this paper
Cite this paper
Bond, J., Hui, S., Schmidt, H. (2001). Linear-Congruence Constructions of Low-Density Parity-Check Codes. In: Marcus, B., Rosenthal, J. (eds) Codes, Systems, and Graphical Models. The IMA Volumes in Mathematics and its Applications, vol 123. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-0165-3_4
Download citation
DOI: https://doi.org/10.1007/978-1-4613-0165-3_4
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-95173-7
Online ISBN: 978-1-4613-0165-3
eBook Packages: Springer Book Archive