  \textit{Matching families} are one of the major ingredients in the
construction of {\em locally decodable codes} (LDCs) and the best known
constructions of LDCs with a constant number of queries are based on matching
families. The determination of the largest size of any matching family in
$\mathbb{Z}_m^n$, where $\mathbb{Z}_m$ is the ring of integers modulo $m$, is
an interesting problem. In this paper, we show an upper bound of
$O((pq)^{0.625n+0.125})$ for the size of any matching family in
$\mathbb{Z}_{pq}^n$, where $p$ and $q$ are two distinct primes. Our bound is
valid when $n$ is a constant, $p\rightarrow \infty$ and $p/q\rightarrow 1$. Our
result improves an upper bound of Dvir {\it et al.}
