  We present a publicly-available code to generate sets of mock Lyman-$\alpha$
(\lya) forest data that have realistic large-scale correlations including those
due to the Baryonic Acoustic Oscillations (BAO). The primary purpose of these
mocks is to test the analysis procedures of the Extended Baryon Oscillation
Survey (eBOSS) and the Dark Energy Spectroscopy Instrument (DESI) surveys. The
transmitted flux fraction, $F(\lambda)$, of background quasars due to \lya\
absorption in the intergalactic medium (IGM) is simulated using the Fluctuating
Gunn-Petterson Approximation (FGPA) applied to Gaussian random fields produced
through the use of fast Fourier transforms (FFT). The output includes the
IGM-\lya\ transmitted flux fraction along quasar lines of sight and a catalog
of high-column-density systems appropriately placed at high-density regions of
the IGM. This output serves as input to additional code that superimposes the
IGM tranmission on realistic quasar spectra, adds absorption by
high-column-density systems and metals, and simulates instrumental transmission
and noise. Redshift space distortions (RSD) of the flux correlations are
implemented by including the large-scale velocity-gradient field in the FGPA
resulting in a correlation function of $F(\lambda)$ that can be accurately
predicted. One hundred realizations have been produced over the 14,000 deg$^2$
DESI survey footprint with 100 quasars per deg$^{2}$. The analysis of these
realizations shows that the correlations of $F(\lambda)$ follows the prediction
within the accuracy of eBOSS survey. The most time-consuming part of the mock
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
