Club filter basis in $\omega_1$

My question is about existing of basis of club filter club($$\omega_1$$) with cardinality $$c$$. Does it exist?

On the one hand, it's consistent with ZFC that $$2^{\aleph_1}=\mathfrak c$$, in which case the whole club filter on $$\aleph_1$$ has cardinality $$\mathfrak c$$.
On the other hand, the continuum hypothesis is consistent with ZFC and implies that the club filter has no basis of size $$\mathfrak c=\aleph_1$$. That's because the diagonal intersection of any $$\aleph_1$$ club sets is again club.