The Completed SDSS-IV extended Baryon Oscillation Spectroscopic Survey:
  Baryon acoustic oscillations with Lyman-$\alpha$ forests
  We present a measurement of baryonic acoustic oscillations (BAO) from
Lyman-$\alpha$ (Ly$\alpha$) absorption and quasars at an effective redshift
$z=2.33$ using the complete extended Baryonic Oscillation Spectroscopic Survey
(eBOSS). The sixteenth and final eBOSS data release (SDSS DR16) contains all
data from eBOSS and its predecessor, the Baryonic Oscillation Spectroscopic
Survey (BOSS), providing $210,005$ quasars with $z_{q}>2.10$ that are used to
measure Ly$\alpha$ absorption. We measure the BAO scale both in the
auto-correlation of Ly$\alpha$ absorption and in its cross correlation with
$341,468$ quasars with redshift $z_{q}>1.77$. Apart from the statistical gain
from new quasars and deeper observations, the main improvements over previous
work come from more accurate modeling of physical and instrumental correlations
and the use of new sets of mock data. Combining the BAO measurement from the
auto- and cross-correlation yields the constraints of the two ratios
$D_{H}(z=2.33)/r_{d} = 8.99 \pm 0.19$ and $D_{M}(z=2.33)/r_{d} = 37.5 \pm 1.1$,
where the error bars are statistical. These results are within $1.5\sigma$ of
the prediction of the flat-$\Lambda$CDM cosmology of Planck~(2016). The
analysis code, \texttt{picca}, the catalog of the flux-transmission field
measurements, and the $\Delta \chi^{2}$ surfaces are publicly available.
