Systematically Measuring Ultra-Diffuse Galaxies (SMUDGes). II. Expanded
  Survey Description and the Stripe 82 Catalog
  We present 226 large ultra-diffuse galaxy (UDG) candidates ($r_e >
5.3$\arcsec, $\mu_{0,g} > 24$ mag arcsec$^{-2}$) in the SDSS Stripe 82 region
recovered using our improved procedure developed in anticipation of processing
the entire Legacy Surveys footprint. The advancements include less constrained
structural parameter fitting, expanded wavelet filtering criteria,
consideration of Galactic dust, estimates of parameter uncertainties and
completeness based on simulated sources, and refinements of our automated
candidate classification. We have a sensitivity $\sim$1 mag fainter in
$\mu_{0,g}$ than the largest published catalog of this region. Using our
completeness-corrected sample, we find that (1) there is no significant decline
in the number of UDG candidates as a function of $\mu_{0,g}$ to the limit of
our survey ($\sim$ 26.5 mag arcsec$^{-2}$); (2) bluer candidates have smaller
S\'ersic $n$; (3) most blue ($g-r < 0.45$ mag) candidates have $\mu_{0,g}
\lesssim 25$ mag arcsec$^{-2}$ and will fade to populate the UDG red sequence
we observe to $\sim 26.5$ mag arcsec$^{-2}$; (4) any red UDGs that exist
significantly below our $\mu_{0,g}$ sensitivity limit are not descended from
blue UDGs in our sample; and (5) candidates with lower $\mu_{0,g}$ tend to
smaller $n$. We anticipate that the final SMUDGes sample will contain $\sim$
30$\times$ as many candidates.
