  We compute explicit bounds in the Gaussian approximation of functionals of
infinite Rademacher sequences. Our tools involve Stein's method, as well as the
use of appropriate discrete Malliavin operators. Although our approach does not
require the classical use of exchangeable pairs, we employ a chaos expansion in
order to construct an explicit exchangeable pair vector for any random variable
which depends on a finite set of Rademacher variables. Among several examples,
which include random variables which depend on infinitely many Rademacher
variables, we provide three main applications: (i) to CLTs for multilinear
forms belonging to a fixed chaos, (ii) to the Gaussian approximation of
weighted infinite 2-runs, and (iii) to the computation of explicit bounds in
CLTs for multiple integrals over sparse sets. This last application provides an
alternate proof (and several refinements) of a recent result by Blei and
Janson.
