The Completed SDSS-IV extended Baryon Oscillation Spectroscopic Survey: Baryon acoustic oscillations with Lyman-$\alpha$ forests

We present a measurement of baryonic acoustic oscillations (BAOs) from Lya absorption and quasars at an effective redshift z = 2.33 using the complete extended Baryonic Oscillation Spectroscopic Survey (eBOSS). The 16th 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 Lya absorption. We measure the BAO scale both in the autocorrelation of Lya 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 +/- 0.19 and D-M (z = 2.33) r(d) = 37.5 +/- 1.1, where the error bars are statistical. These results are within 1.5s of the prediction of the flat-.CDM cosmology of Planck (2016). The analysis code, picca, the catalog of the flux transmission field measurements, and the Delta chi(2) surfaces are publicly available.