  We present a catalog of 5598 ultra-diffuse galaxy (UDG) candidates with
effective radius $r_e > 5.3$ arcsec distributed throughout the southern portion
of the DESI Legacy Imaging Survey covering $\sim$ 15000 deg$^2$. The catalog is
most complete for physically large ($r_e > 2.5$ kpc) UDGs lying in the redshift
range $1800 \lesssim cz/{\rm km\ s}^{-1} \lesssim 7000$, where the lower bound
is defined by where incompleteness becomes significant for large objects on the
sky and the upper bound by our minimum angular size selection criterion.
Because physical size is integral to the definition of a UDG, we develop a
method {of distance estimation} using existing redshift surveys. With three
different galaxy samples, two of which contain UDGs with spectroscopic
redshifts, we estimate that the method has a redshift accuracy of $\sim$ 75%
when the method converges, although larger, more representative spectroscopic
UDG samples are needed to fully understand the behavior of the method. We are
able to estimate distances for 1079 of our UDG candidates (19%). Finally, to
illustrate uses of the catalog, we present distance independent and dependent
results. In the latter category we establish that the red sequence of UDGs lies
on the extrapolation of the red sequence relation for bright ellipticals and
that the environment-color relation is at least qualitatively similar to that
of high surface brightness galaxies. Both of these results challenge some of
the models proposed for UDG evolution.
