Reduction modulo 2 and 3 of Euclidean lattices
For a lattice \(L\) in \(n\)-dimensional Euclidean space, shortest representatives of the congruence classes modulo \(dL\) are studied for \(d=2\) and \(d=3\). Let \(m\) and \(m'\) be the minimal norm and the norm of the second layer of \(L\). Then it is shown in particular that for \(d=2\) (resp., \(d=3\)) one must consider vectors of norm up to \(2m\) (resp., up to \(2m+m'\)) in order to represent all non-zero classes. Furthermore the structure of the sets of such vectors having a fixed norm and representing a fixed class is discussed. The methods and results may be viewed as a generalization of arguments used, with \(d=2\), by J. H. Conway in his famous characterization of the Leech lattice. Here also the Leech lattice modulo 3 is studied, and an application is given to the classification of sublattices with cyclic quotients of order \(d\).
scientific article; zbMATH DE number 1846026