  We consider the algorithmic decision problem that takes as input an
$n$-vertex $k$-uniform hypergraph $H$ with minimum codegree at least $m-c$ and
decides whether it has a matching of size $m$. We show that this decision
problem is fixed parameter tractable with respect to $c$. Furthermore, our
algorithm not only decides the problem, but actually either finds a matching of
size $m$ or a certificate that no such matching exists. In particular, when
$m=n/k$ and $c=O(\log n)$, this gives a polynomial-time algorithm, that given
any $n$-vertex $k$-uniform hypergraph $H$ with minimum codegree at least
$n/k-c$, finds either a perfect matching in $H$ or a certificate that no
perfect matching exists.
