Perfect lattice

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title: "Perfect lattice" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["quadratic-forms"] description: "Euclidean lattice" topic_path: "general/quadratic-forms" source: "https://en.wikipedia.org/wiki/Perfect_lattice" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Euclidean lattice ::

In mathematics, a perfect lattice (or perfect form) is a lattice in a Euclidean vector space, that is completely determined by the set S of its minimal vectors in the sense that there is only one positive definite quadratic form taking value 1 at all points of S. Perfect lattices were introduced by . A strongly perfect lattice is one whose minimal vectors form a spherical 4-design. This notion was introduced by .

proved that a lattice is extreme if and only if it is both perfect and eutactic.

The number of perfect lattices in dimensions 1, 2, 3, 4, 5, 6, 7, 8 is given by 1, 1, 1, 2, 3, 7, 33, 10916 . summarize the properties of perfect lattices of dimension up to 7. verified that the list of 10916 perfect lattices in dimension 8 found by Martinet and others is complete. It was proven by that only 2408 of these 10916 perfect lattices in dimension 8 are actually extreme lattices.

References

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