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 ≤ n ≤ 16, our bound is tighter than the three-point bound of Cohn, de Laat and Salmon. For n ∈ [17, 23] ∪ [25, 40] ∪ [43, 57] ∪ [62, 75]∪ [83, 91], our bound lies below the published numerical values of the Cohn--Elkies bound.