  This is the first in a series of papers on the properties of ultra-diffuse
galaxies (UDGs) in clusters of galaxies. We present an updated catalog of UDGs
in the Coma cluster using \textit{g}- and \textit{r}-band images obtained with
Hyper Suprime-Cam (HSC) of the Subaru telescope. We develop a method to find
UDGs even in the presence of contaminating objects, such as halos and
background galaxies. This study expands upon our previous works that covered
about half the area of the Coma cluster. The HSC observations covered the whole
Coma cluster up to the virial radius and beyond (an area twice larger than the
previous studies) and doubled the numbers of UDGs ($r_{\rm eff, r} \geq 1.5$
kpc) and sub-UDGs ($1.0 \leq r_{\rm eff, r} < 1.5$ kpc) to 774 and 729
respectively. The new UDGs show internal properties consistent with those of
the previous studies (e.g., S\'ersic index of approximately 1), and are
distributed across the cluster, with a concentration around the cluster center.
The whole cluster coverage clearly revealed an excess of their distribution
toward the east to south-west direction along the cluster center, where Coma
connects to the large scale structure, and where a known substructure exists
(the NGC4839 subgroup). The alignment of the UDG distribution along the large
scale structure around Coma supports the interpretation that most of them lie
at the distance of the Coma cluster and the NGC4839 subgroup.
