Improved lattice-packing bounds in small dimensions

We strengthen the upper bound on the density of a lattice packing in small dimensions. A lattice cannot have too many short vectors; we modify the Cohn--Elkies linear program accordingly by allowing the auxiliary function to be positive on a short interval beyond the minimum distance. For $11 \leq n \leq 16$, our bound is tighter than the three-point bound of Cohn, de Laat and Salmon. For $n \in [17, 23] \cup [25, 40] \cup [43, 57] \cup [62, 75]\cup [83, 91]$, our bound lies below the published numerical values of the Cohn--Elkies bound.