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International Symposium on Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes

AAECC 1999: Applied Algebra, Algebraic Algorithms and Error-Correcting Codes pp 104–120Cite as

An Algorithm to Compute a Nearest Point in the Lattice A * n

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Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1719))

Abstract

The lattice A * n is an important lattice because of its covering properties in low dimensions. Conway and Sloane [3] appear to have been the first to consider the problem of computing the nearest lattice point in A * n . They developed and later improved [4] an algorithm which is able to compute a nearest point in O(n 2) arithmetic steps. In this paper, a new algorithm is developed which is able to compute a nearest point in O(n log n) steps.

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References

  1. I. Vaughan L. Clarkson. Approximation of Linear Forms by Lattice Points with Applications to Signal Processing. PhD thesis, The Australian National University, 1997.

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  2. I. Vaughan L. Clarkson, Stephen D. Howard, and Iven M. Y. Mareels. Parameter estimation and association from pulse time-of-arrival data. Submitted to Signal Process., 1997.

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  3. J. H. Conway and N. J. A. Sloane. Fast quantizing and decoding algorithms for lattice quantizers and codes. IEEE Trans. Inform. Theory, IT-28(2):227–232, March 1982.

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  4. J. H. Conway and N. J. A. Sloane. Soft decoding techniques for codes and lattices, including the Golay code and the Leech lattice. IEEE Trans. Inform. Theory, IT-32(1):41–50, January 1986.

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  5. J. H. Conway and N. J. A. Sloane. Sphere Packings, Lattices and Groups. Springer-Verlag, Berlin, 1988.

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  6. Donald E. Knuth. The Art of Computer Programming, volume 3 (Sorting and Searching). Addison-Wesley, Reading, Ma., 1973.

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  7. P. van Emde Boas. Another NP-complete partition problem and the complexity of computing short vectors in a lattice. Technical Report 81-04, Mathematisch Instituut, Roetersstraat 15, 1018 WB Amsterdam, The Netherlands, April 1981.

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Author information

Authors and Affiliations

  1. Department of Electrical and Electronic Engineering, The University of Melbourne, Parkville, Victoria, 3052, Australia

    I. Vaughan & L. Clarkson

Authors
  1. I. Vaughan
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  2. L. Clarkson
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Editor information

Editors and Affiliations

  1. Department of Electrical Engineering, University of Hawaii, 2540 Dole St., # 483, Honolulu, HI, 96822, USA

    Marc Fossorier  & Shu Lin  & 

  2. Institute of Industrial Science, University of Tokyo, 7-22-1, Roppongi, Minato-ku, Tokyo, 106, Japan

    Hideki Imai

  3. AAECC/IRIT, Université Paul Sabatier, 118, Route de Narbonne, F31067, Toulouse, Cedex, France

    Alain Poli

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© 1999 Springer-Verlag Berlin Heidelberg

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Vaughan, I., Clarkson, L. (1999). An Algorithm to Compute a Nearest Point in the Lattice A * n . In: Fossorier, M., Imai, H., Lin, S., Poli, A. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 1999. Lecture Notes in Computer Science, vol 1719. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-46796-3_11

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  • DOI: https://doi.org/10.1007/3-540-46796-3_11

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-66723-0

  • Online ISBN: 978-3-540-46796-0

  • eBook Packages: Springer Book Archive

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