A new upper bound for the minimum of an integral lattice of determinant 1

Abstract

Let Λ be an n-dimensional integral lattice of determinant 1. We show that, for all sufficiently large n, the minimal nonzero squared length in Λ does not exceed [ ( n + 6 ) /10 ]. This bound is a consequence of some new conditions on the theta series of these lattices; these conditions also enable us to find the greatest possible minimal squared length in all dimensions n ≤ 33. In particular we settle the ‘‘no-roots’’ problem: there is a determinant 1 lattice containing no vectors of squared length 1 or 2 precisely when n ♢ 23, n © 25. There are also analogues of all these results for codes.