A tomographic map of the large-scale matter distribution using the eBOSS
  Stripe 82 Ly-$\alpha$ forest
  The Lyman-$\alpha$ (hereafter Ly-$\alpha$) forest is a probe of large-scale
matter density fluctuations at high redshift, $z > 2.1$. It consists of HI
absorption spectra along individual lines-of-sight. If the line-of-sight
density is large enough, 3D maps of HI absorption can be inferred by
tomographic reconstruction. In this article, we investigate the Ly-$\alpha$
forest available in the Stripe 82 field ($220\,\mathrm{deg^{2}}$), based on the
quasar spectra from SDSS Data Release DR16. The density of observed quasar
spectra is $37\,\mathrm{deg^{-2}}$ with a mean pixel signal-to-noise ratio of
two per angstrom. This study provides an intermediate case between the average
SDSS density and that of the much denser but smaller CLAMATO survey. We derive
a 3D map of large-scale matter fluctuations from these data, using a Wiener
filter technique. The total volume of the map is $0.94\,\mathrm{h^{-3}
Gpc^{3}}$. Its resolution is $13\,\mathrm{h^{-1} Mpc}$, which is related to the
mean transverse distance between nearest lines-of-sight. From this map, we
provide a catalog of voids and protocluster candidates in the cosmic web. The
map-making and void catalog are compared to simulated eBOSS Stripe 82
observations. A stack over quasar positions provides a visualization of the
Ly-$\alpha$ quasar cross-correlation. This tomographic reconstruction
constitutes the largest-volume high-redshift 3D map of matter fluctuations.
