# Upper bound on the number of shortest vectors in a lattice

Given a Lattice $L$ (e.g. by its Gram-Matrix or via a basis) I would like to know whether there is an upper bound on the number of shortest vectors in $L$. Available information on $L$ includes dimension, rank as well as an upper and a lower bound on the minimum of $L$.

The question arises in the following context: The group of automorphisms of $L$ ($Aut(L)$) acts on the set of shortest vectors of $L$ and an upper bound on the cardinality of this set would yield some information on which stabilizers are to be expected.

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See this earlier MO question: "Estimating the number of short vectors in a lattice": mathoverflow.net/questions/71052 – Joseph O'Rourke Sep 14 '11 at 13:23
Also see SPLAG (= Sphere Packings, Lattices, and Groups by Conway and Sloane) about "kissing numbers". [The acronym "SPLAG", though risible, is not disrespectful since the authors use it themselves.] – Noam D. Elkies Sep 14 '11 at 14:32
Thanks a lot. I somehow missed the quite obvious connection between my question and the kissing number. Sadly there does not seem to be a naive upper bound and the number of results in dimension 248 is rather limited. – Sebastian Schoennenbeck Sep 15 '11 at 7:12