## The strongly perfect lattices of dimension 10.(English)Zbl 0997.11049

By a theorem of the second author, a lattice in Euclidean $$n$$-space whose minimal vectors form a spherical 4-design (resp., 2-design) is perfect (resp., eutactic) in the sense of Voronoi. A lattice having this 4-design property is called strongly perfect (its density then attains a local maximum). The classification of such lattices was known for $$n\leq 8$$, where only root lattices and their duals occur, and also for $$n=9$$ and 11 where the second author had shown that no examples exist. For $$n=10$$, so far two less trivial examples were known, an integral lattice with minimum 4 and its dual, and their kissing numbers (270 and 240, respectively) were known to be the only possible ones. In a rather involved analysis it is verified now that, up to similarity, no other strongly perfect lattices exist in dimension 10.

### MSC:

 11H31 Lattice packing and covering (number-theoretic aspects) 05B30 Other designs, configurations 11H06 Lattices and convex bodies (number-theoretic aspects)

### Keywords:

perfect lattice; spherical design

Magma
Full Text:

### References:

 [1] Bachoc, C., Venkov, B., Modular forms, lattices and spherical designs. In [EM]. · Zbl 1061.11035 [2] Cassels, J.W.S., Rational quadratic forms. Academic Press (1978). · Zbl 0395.10029 [3] Conway, J.H., Sloane, N.J A., Sphere Packings, Lattices and Groups. 3rd edition, Springer-Verlag (1998). · Zbl 0915.52003 [4] Conway, J.H., Sloane, N.J.A., On Lattices Equivalent to Their Duals. J. Number Theory48 (1994), 373-382. · Zbl 0810.11041 [5] Réseaux euclidiens, designs sphériques et groupes. Edited by J. Martinet. Enseignement des Mathématiques, monographie 37, to appear. · Zbl 1054.11034 [6] The Magma Computational Algebra System for Algebra, Number Theory and Geometry. available via the magma home page http://wvw. maths. usyd. edu. au:8000/u/magma/. [7] Martinet, J., Les Réseaux parfaits des espaces Euclidiens. Masson (1996). · Zbl 0869.11056 [8] Martinet, J., Sur certains designs sphériques liés à des réseaux entiers. In [EM]. · Zbl 1065.11050 [9] Milnor, J., Husemoller, D., Symmetric bilinear forms. Springer-Verlag (1973). · Zbl 0292.10016 [10] Scharlau, W., Quadratic and Hermitian Forms. Springer Grundlehren270 (1985). · Zbl 0584.10010 [11] Souvignier, B., Irreducible finite integral matrix groups of degree 8 and 10. Math. Comp.61207 (1994), 335-350. · Zbl 0830.20074 [12] Venkov, B., Réseaux et designs sphériques. Notes taken by J. Martinet of lectures by B. Venkov at Bordeaux (1996/1997). In [EM].
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.