  We carry out a numerical study of the spinless modular bootstrap for
conformal field theories with current algebra $U(1)^c \times U(1)^c$, or
equivalently the linear programming bound for sphere packing in $2c$
dimensions. We give a more detailed picture of the behavior for finite $c$ than
was previously available, and we extrapolate as $c \to \infty$. Our
extrapolation indicates an exponential improvement for sphere packing density
bounds in high dimensions. Furthermore, we study when these bounds can be
tight. Besides the known cases $c=1/2$, $4$, and $12$ and the conjectured case
$c=1$, our calculations numerically rule out sharp bounds for all other $c<90$,
by combining the modular bootstrap with linear programming bounds for spherical
codes.
