  Synthetic data sets are used in cosmology to test analysis procedures, to
verify that systematic errors are well understood and to demonstrate that
measurements are unbiased. In this work we describe the methods used to
generate synthetic datasets of Lyman-$\alpha$ quasar spectra aimed for studies
with the Dark Energy Spectroscopic Instrument (DESI). In particular, we focus
on demonstrating that our simulations reproduces important features of real
samples, making them suitable to test the analysis methods to be used in DESI
and to place limits on systematic effects on measurements of Baryon Acoustic
Oscillations (BAO). We present a set of mocks that reproduce the statistical
properties of the DESI early data set with good agreement. Additionally, we use
full survey synthetic data to forecast the BAO scale constraining power with
DESI.
