  We extend Bourgain's $L^2$-wellposedness result for the KP-II equation on
$\mathbb{T}^2$ to initial data with negative Sobolev regularity. The key
ingredient is a new linear $L^4$-Strichartz estimate which is effective on
frequency-dependent time scales. The $L^4$-Strichartz estimates follow from
combining an $\ell^2$-decoupling inequality recently proved by
Guth--Maldague--Oh with semiclassical Strichartz estimates. Moreover, we rely
on a variant of Bourgain's bilinear Strichartz estimate on frequency-dependent
times, which is proved via the C\'ordoba--Fefferman square function estimate.
