Hostname: page-component-7dd5485656-kp629 Total loading time: 0 Render date: 2025-10-31T13:06:01.815Z Has data issue: false hasContentIssue false
  • English
  • Français

Double circulant constructions of the Leech lattice

Part of: Theory of error-correcting codes and error-detecting codes Geometry of numbers

Published online by Cambridge University Press:  09 April 2009

Robin Chapman
Robin Chapman
Affiliation:
School of Mathematical Sciences University of ExeterExeter EX4 4QEUK e-mail: rjc@maths.ex.ac.uk
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider the problem of finding, for each even number m, a basis of orthogonal vectors of length in the Leech lattice. We give such a construction by means of double circulant codes whenever m = 2p and p is a prime not equal to 11. From this one can derive a construction for all even m not of the form 2· 11r.

MSC classification

Secondary: 11H06: Lattices and convex bodies 11H71: Relations with coding theory 94B05: Linear codes, general

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 69 , Issue 3 , December 2000 , pp. 287 - 297
Copyright
Copyright © Australian Mathematical Society 2000

References

[1]Bonnecaze, A., Solé, P. and Calderbank, A. R., ‘Quaternary quadratic residue codes and unimodular lattices’, IEEE Trans. Inform. Theory 41 (1985), 366–377.CrossRefGoogle Scholar
[2]Calderbank, A. R. and Sloane, N. J. A., ‘Double circulant codes over 4 and even unimodular lattices’, J. Algebraic Combin. 6 (1997), 119–131.CrossRefGoogle Scholar
[3]Conway, J. H., ‘A characterisation of Leech's lattice’, Invent. Math. 7 (1969), 137–142.CrossRefGoogle Scholar
[4]Ebeling, W., Lattices and codes (Vieweg, Braunschweig/Wiesbaden, 1994).CrossRefGoogle Scholar
[5]Harada, M., Solé, P. and Gaborit, P., ‘Self-dual codes over 4 and unimodular lattices: a survey’, in: Algebra and Combinatorics: an International Congress, ICAC '97, Hong Kong (Springer-Verlag, Singapore, 1999).Google Scholar
[6]Knapp, A., Elliptic curves (Princeton University Press, Princeton, 1992).Google Scholar
[7]Leech, J., ‘Notes on sphere packings’, Canad. J. Math 19 (1967), 251–267.CrossRefGoogle Scholar
[8]McKay, J., ‘A setting for the Leech lattice’ in: Finite Groups '72 (ed. Gagen, T. M.) (North-Holland, Amsterdam, 1973).Google Scholar
[9]Schoeneberg, B., Elliptic modular functions (Springer, Berlin, 1974).CrossRefGoogle Scholar