Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 44 (2003), No. 1, 229-234.

Even Lattices with Covering Radius < {√ \atop 2} Gabriele Nebe Abteilung Reine Mathematik, Universität Ulm, 89069 Ulm, Germany e-mail: nebe@mathematik.uni-ulm.de 1. Introduction Let L be a lattice in Euclidean space V := ℝ ⊗ L. Then the covering radius of L is the smallest number r ∈ ℝ such that the spheres with radius r around all lattice points cover the whole space V.

The famous Leech lattice Λ_{24}, the unique even unimodular lattice of rank 24 with minimal distance 4, has covering radius \sqrt{2}, as shown in [2] (see [3, Chapter 23]). This is the main observation that enables Conway to calculate the automorphism group of the 26-dimensional

even unimodular Lorentzian lattice Λ_{24} ⊥ ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) ([1], see [3, Chapter 27]). The present

article is motivated by a question of Richard Parker, who wants to have a list of all even lattices L with covering radius ≤ \sqrt{2} to construct examples of Lorentzian lattices L ⊥

( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) for which he can calculate the automorphism group. As a first step, the even

lattices of covering radius < √ ; \overline{2} are classified in this note.

If L is an even lattice with covering radius < \sqrt{2}, then for every v ∈ V, there is a vector l ∈ L with (v − l, v − l) < 2, where (v, w) denotes the scalar product of two vectors v, w ∈ V. In particular if v = \frac{1}{2}w with w ∈ L, then (w − 2l, w − 2l) < 8. Since L is even, this means that every coset in L/2L contains a vector of square length ≤ 6. Let μ(L) denote the minimal m such that every coset in L/2L contains a vector of norm ≤ m. The easy but crucial observation is stated in Lemma 1: If L is an even lattice with μ(L) ≤ 6, then every norm 8 vector in L gives rise to a norm 2 vector in L which enables to classify these lattices according to the sublattices spanned by the vectors of norm 2 in L. 32 of the lattices L with μ(L) ≤ 6 are root lattices (Theorem 6), where the largest dimension is 10, achieved by E_{8}A_{2}. For the other 51 lattices (given in Theorem 7) the root sublattice is not of full rank. This list

0138-4821/93 $ 2.50 \begin{matrix}c \\ ⊗ \end{matrix} 2003 Heldermann Verlag