  We present an updated general purpose nuclear equation of state (EoS) for use
in simulations of core-collapse supernovae, neutron star mergers and black hole
collapse. This EoS is formulated in the context of Density Functional Theory
(DFT) and is generalized to include all DFT EoSs consistent with known nuclear
and astrophysical constraints. This EoS also allows for the possibility of the
formation of material with a net proton excess ($Y_p > 0.5$) and has an
improved treatment of the nuclear statistical equilibrium and the transition to
heavy nuclei as the density approaches nuclear matter density. We include the
effects of pions in the regime above nuclear matter density and incorporate all
of the known mesonic and baryonic states at high temperature.
  We analyze how a 3-body nuclear force term in the DFT at high densities
stiffens the EoS to satisfy the maximum neutron star constraint, however the
density dependence of the symmetry anergy and the formation of pions at high
temperatures allows for a softening of the central core in supernova collapse
calculations leading to a robust explosion. We also add the possibility of a
transition to a QCD chiral-symmetry-restoration and deconfinement phase at
densities above nuclear matter density. This paper details the physics, and
constraints on, this new EoS and presents an illustration of its implementation
in both neutron stars and core-collapse supernova simulations. We present the
first results from core-collapse supernova simulations with this EoS.
