  Inspired by the lattice structure of the unhydrated variant of the
superconducting material Na$_x$CoO$_2 \cdot$yH$_2$O at $ x = {1/3}$, we study
the $t$-$J$ model on a honeycomb lattice by using an efficient loop-cluster
algorithm. The low-energy physics of the undoped system and of the single hole
sector is described by a systematic low-energy effective field theory. The
staggered magnetization per spin $\widetilde{{\cal M}}_s = 0.2688(3)$, the spin
stiffness $\rho_s = 0.102(2) J$, the spin wave velocity $c= 1.297(16) J a$, and
the kinetic mass $M'$ of a hole are obtained by fitting the numerical Monte
Carlo data to the effective theory predictions.
