  The q-state Potts antiferromagnet on a lattice $\Lambda$ exhibits nonzero
ground state entropy $S_0=k_B \ln W$ for sufficiently large q and hence is an
exception to the third law of thermodynamics. An outstanding challenge has been
the calculation of W(sq,q) on the square (sq) lattice. We present here an exact
calculation of W on an infinite-length cyclic strip of the square lattice which
embodies the expected analytic properties of W(sq,q). Similar results are given
for the kagom\'e lattice.
