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Low density parity check codes based on sparse matrices with no small cycles

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Crytography and Coding (Cryptography and Coding 1997)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1355))

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Abstract

In this paper we give a systematic construction of matrices with constant row weights and column weights and arbitrarily large girths. This resolves a problem raised by D. MacKay. The matrices are used in the generator matrices of linear codes. We give the experiment performance results for codes whose associated matrices have girth 8. We also give a randomized construction of matrices with constant row sums and column sums and few 4-cycles. The codes generated using the matrices are used to encode bit streams for a Gaussian channel and decoded using a decoding algorithm that combines features of the algorithms given by MacKay and Cheng and McEliece. The experimental performance results for codes generated using the random matrices are compared to those of the systematically constructed codes. The results show that the codes generated using the random codes with smaller block sizes perform as well as the systematic codes with bigger block sizes. The performance of the systematic codes, for specified weights, can be used to tailor the random codes. MATLAB routines for the construction for the girth 8 case and a special girth 4 case are included.

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References

  1. J. F. Cheng and R. J. McEliece, Some high rate near capacity codes for the Gaussian channel, Preprint.

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  2. R. G. Gallager, Low density parity check codes, IRE Transactions on Information Theory, Jan. 1962.

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  3. R. G. Gallager, Low Density Parity Check Codes, MIT Press, 1963.

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  4. D. J. C. MacKay, Good error-correcting codes based on very sparse matrices, Preprint.

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  5. D. J. C. MacKay and C. P. Hesketh, Sensitivity of low density parity check codes to decoding assumptions, Preprint.

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  6. J. Pearl, Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference, Morgan Kaufmann Publishers, 1988.

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Author information

Authors and Affiliations

  1. Science Applications International Corp, 4015 Hancock Street, 92110, San Diego, CA

    J. Bond

  2. Department of Mathematical Sciences, San Diego State University, 92182, San Diego, CA

    S. Hui

  3. Technology Service Corporation, 962 Wayne Avenue, Suite 800, 20910, Silver Spring, MD

    H. Schmidt

Authors
  1. J. Bond
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  2. S. Hui
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  3. H. Schmidt
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Editor information

Michael Darnell

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© 1997 Springer-Verlag Berlin Heidelberg

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Bond, J., Hui, S., Schmidt, H. (1997). Low density parity check codes based on sparse matrices with no small cycles. In: Darnell, M. (eds) Crytography and Coding. Cryptography and Coding 1997. Lecture Notes in Computer Science, vol 1355. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0024448

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  • DOI: https://doi.org/10.1007/BFb0024448

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-63927-5

  • Online ISBN: 978-3-540-69668-1

  • eBook Packages: Springer Book Archive

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