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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)

Software:

Magma
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References:

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