  We perform lattice Monte Carlo simulations for up to 66 unitary fermions in a
finite box using a highly improved lattice action for nonrelativistic spin 1/2
fermions. We obtain a value of $0.366^{+0.016}_{-0.011}$ for the Bertsch
parameter, defined as the energy of the unitary Fermi gas measured in units of
the free gas energy in the thermodynamic limit. In addition, for up to four
unitary fermions, we compute the spectrum of the lattice theory by exact
diagonalization of the transfer matrix projected onto irreducible
representations of the octahedral group for small to moderate size lattices,
providing an independent check of our few-body simulation results. We compare
our exact numerical and simulation results for the spectrum to benchmark
studies of other research groups, as well as perform an extended analysis of
our lattice action improvement scheme, including an analysis of the errors
associated with higher partial waves and finite temporal discretization.
