  Caching is a promising solution to satisfy the ever increasing demands for
the multi-media traffics. In caching networks, coded caching is a recently
proposed technique that achieves significant performance gains over the uncoded
caching schemes. However, to implement the coded caching schemes, each file has
to be split into $F$ packets, which usually increases exponentially with the
number of users $K$. Thus, designing caching schemes that decrease the order of
$F$ is meaningful for practical implementations. In this paper, by reviewing
the Ali-Niesen caching scheme, the placement delivery array (PDA) design
problem is firstly formulated to characterize the placement issue and the
delivery issue with a single array. Moreover, we show that, through designing
appropriate PDA, new centralized coded caching schemes can be discovered.
Secondly, it is shown that the Ali-Niesen scheme corresponds to a special class
of PDA, which realizes the best coding gain with the least $F$. Thirdly, we
present a new construction of PDA for the centralized caching system, wherein
the cache size of each user $M$ (identical cache size is assumed at all users)
and the number of files $N$ satisfies $M/N=1/q$ or ${(q-1)}/{q}$ ($q$ is an
integer such that $q\geq 2$). The new construction can decrease the required
$F$ from the order $O\left(e^{K\cdot\left(\frac{M}{N}\ln \frac{N}{M}
+(1-\frac{M}{N})\ln \frac{N}{N-M}\right)}\right)$ of Ali-Niesen scheme to
$O\left(e^{K\cdot\frac{M}{N}\ln \frac{N}{M}}\right)$ or
$O\left(e^{K\cdot(1-\frac{M}{N})\ln\frac{N}{N-M}}\right)$ respectively, while
the coding gain loss is only $1$.
