  This work focuses on non-adaptive group testing, with a primary goal of
efficiently identifying a set of at most $d$ defective elements among a given
set of elements using the fewest possible number of tests. Non-adaptive
combinatorial group testing often employs disjunctive codes and union-free
codes. This paper discusses union-free codes with fast decoding (UFFD codes), a
recently introduced class of union-free codes that combine the best of both
worlds -- the linear complexity decoding of disjunctive codes and the fewest
number of tests of union-free codes. In our study, we distinguish two
subclasses of these codes -- one subclass, denoted as $(=d)$-UFFD codes, can be
used when the number of defectives $d$ is a priori known, whereas $(\le
d)$-UFFD codes works for any subset of at most $d$ defectives. Previous studies
have established a lower bound on the rate of these codes for $d=2$. Our
contribution lies in deriving new lower bounds on the rate for both $(=d)$- and
$(\le d)$-UFFD codes for an arbitrary number $d \ge 2$ of defectives. Our
results show that for $d\to\infty$, the rate of $(=d)$-UFFD codes is twice as
large as the best-known lower bound on the rate of $d$-disjunctive codes. In
addition, the rate of $(\le d)$-UFFD code is shown to be better than the known
lower bound on the rate of $d$-disjunctive codes for small values of $d$.
