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.