  In this note, we prove that there exists a universal constant
$c=\frac{43}{50}$ such that for every $k\in \mathbb{N}$ and every $d<k/2$,
every $k$-uniform hypergraph on $n$ vertices and with minimum $d$-degree at
least $(c+o_n(1))\binom{n-d}{k-d}$ contains a perfect matching. This is the
first such bound which is independent of $k$, and therefore, improves all
previously known bounds when $k$ is large. Our approach is based on combining
the seminal work of Alon et al. with known bounds on a conjectured
probabilistic inequality due to Feige.
