  LiGa$_5$O$_8$ in the spinel type structure is investigated as a potential
ultra-wide-band-gap semiconductor. The band structure is determined using the
quasiparticle self-consistent $GW$ method and the optical properties are
calculated at the Bethe Salpeter Equation level including electron-hole
interaction effects. The optical gap including exciton effects and an estimate
of the zero-point motion electron phonon coupling renormalizations is estimated
to be about 5.2$\pm0.1$ eV with an exciton binding energy of about 0.4 eV. Si
doping as potential n-type dopant is investigated and found to be a promising
shallow donor.
