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Tables of integral unimodular lattices constructed as $$k$$-neighbors in $$\mathbb{Z}^n$$. (Tables de réseaux entiers unimodulaires construits comme $$k$$-voisins de $$\mathbb{Z}^n$$.) (French) Zbl 0906.11033
Two lattices $$L$$ and $$M$$ on euclidean $${\mathbb R}^n$$ are said to be $$k$$-neighbors for some positive integer $$k$$ if both $$L/(L\cap M)$$ and $$M/(L\cap M)$$ are cyclic groups of order $$k$$. The integral $$k$$-neighbors of the standard lattice $${\mathbb Z}^n$$ represent all classes of self-dual lattices (the proof is sketched). In the tables published here one can find such a description for any class up to $$n=24$$, together with some information related to the enumeration of these lattices in [J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups. 2nd ed. (1993; Zbl 0785.11036)].

##### MSC:
 11H06 Lattices and convex bodies (number-theoretic aspects) 11E12 Quadratic forms over global rings and fields 11-04 Software, source code, etc. for problems pertaining to number theory
##### Keywords:
unimodular lattice; neighbor lattice
Zbl 0785.11036
Full Text:
##### References:
 [1] Bacher, R., Réseaux unimodulaires sans automorphismes, Thèse No 2597, Université de Genève, 1993. [2] Conway, J.H. et Sloane, N.J.A., Sphere Packings, Lattices and Groups, Springer, 1988. · Zbl 0634.52002 [3] Kneser, M., Klassenzahlen definiter quadratischer Formen, Arch. der Math. vol.VIII (1957), 241-250. · Zbl 0078.03801 [4] Mimura, Y., Explicit examples of unimodular lattices with the trivial automorphism group, Proceeding of KAIST Mathematics Workshop, vol.5, Algebra and topology, Korea Adv.Inst.Sci.Tech., Taejon, 91-95. · Zbl 0735.11026 [5] Schulze-Pillot, R., An algorithm for computing genera of ternary and quaternary quadratic forms, Proceedings of the International Symposium on Symbolic and Algebraic Computation, Bonn (1991), 134-143. · Zbl 0925.11045
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