Mock data sets for the Eboss and DESI Lyman-$\alpha$ forest surveys
{We present a publicly-available code to generate mock Lyman-$\alpha$ (\lya) forest data sets. The code is based on the Fluctuating Gunn-Peterson Approximation (FGPA) applied to Gaussian random fields and on the use of fast Fourier transforms (FFT). The output includes spectra of lya transmitted flux fraction, $F$, a quasar catalog, and a catalog of high-column-density systems. While these three elements have realistic correlations, additional code is then used to generate realistic quasar spectra, to add absorption by high-column-density systems and metals, and to simulate instrumental effects. Redshift space distortions (RSD) are implemented by including the large-scale velocity-gradient field in the FGPA resulting in a correlation function of $F$ that can be accurately predicted. One hundred realizations have been produced over the 14,000 deg$^2$ Dark Energy Spectroscopy Instrument (DESI) survey footprint with 100 quasars per deg$^{2}$, and they are being used for the Extended Baryon Oscillation Survey (eBOSS) and DESI surveys. The analysis of these realizations shows that the correlation of $F$ follows the prediction within the accuracy of eBOSS survey. The most time-consuming part of the production occurs before application of the FGPA, and the existing pre-FGPA forests can be used to easily produce new mock sets with modified redshift-dependent bias parameters or observational conditions.