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A lattice without a basis of minimal vectors

Part of: Geometry of numbers

Published online by Cambridge University Press:  26 February 2010

J. H. Conway  and
N. J. A. Sloane
J. H. Conway
Affiliation:
Department of Mathematics, Princeton University, Princeton, New Jersey 08544, U.S.A.
N. J. A. Sloane
Affiliation:
Mathematical Sciences Research Center, AT&T Bell Laboratories, Murray Hill, New Jersey 07974, U.S.A.
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Abstract

It is shown that in all dimensions n ≥ 11 there exists a lattice which is generated by its minimal vectors but in which no set of n minimal vectors forms a basis.

MSC classification

Secondary: 11H99: None of the above, but in this section
Type
Research Article
Information
Mathematika , Volume 42 , Issue 1 , June 1995 , pp. 175 - 177
Copyright
Copyright © University College London 1995

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References

1.Conway, J. H. and Sloane, N. J. A.. Sphere Packings, Lattices and Groups, Second edition (Springer-Verlag, New York, 1993).CrossRefGoogle Scholar
2.Csóka, G.. There exists a basis of minimal vectors in every n ≤ 7 dimensional perfect lattice. Ann. Univ. Sci. Budapest. Eotvos Sect. Math., 30 (1987), 245–258.Google Scholar
3.Engel, P.. Geometric crystallography. In Handbook of Convex Geometry, Vol. B (edited by Gruber, P. M. and Wills, J. M.) (North-Holland, Amsterdam, 1993), pp. 989–1041.CrossRefGoogle Scholar
4.Martinet, J.. Les Réseaux Parfaits des Espaces Euclidiens (preprint, 1992).Google Scholar
5.Ryskov, S. S.. On the problem of the determination of the perfect quadratic forms in many variables. Trudy Mat. Inst. Steklov. Moscow, 142 (1976), 215–239. English translation in Proc. Steklov. Inst. Math., 142 (1976), 233–259.Google Scholar
6.Senechal, M.. Introduction to lattice geometry. In From Number Theory to Physics (edited by Waldschmidt, M.et al.) (Springer-Verlag, New York, 1992), pp. 476–495.CrossRefGoogle Scholar