  In caching system, it is desirable to design a coded caching scheme with the
transmission load $R$ and subpacketization $F$ as small as possible, in order
to improve efficiency of transmission in the peak traffic times and to decrease
implementation complexity. Yan et al. reformulated the centralized coded
caching scheme as designing a corresponding $F\times K$ array called placement
delivery array (PDA), where $F$ is the subpacketization and $K$ is the number
of users. Motivated by several constructions of PDAs, we introduce a framework
for constructing PDAs, where each row is indexed by a row vector of some matrix
called row index matrix and each column's index is labelled by an element of a
direct product set. Using this framework, a new scheme is obtained, which can
be regarded as a generalization of some previously known schemes. When $K$ is
equal to ${m\choose t}q^t$ for positive integers $m$, $t$ with $t<m$ and $q\geq
2$, we show that the row index matrix must be an orthogonal array if all the
users have the same memory size. Furthermore, the row index matrix must be a
covering array if the coded gain is ${m\choose t}$, which is the maximal coded
gain under our framework. Consequently the lower bounds on the transmission
load and subpacketization of the schemes are derived under our framework.
Finally, using orthogonal arrays as the row index matrix, we obtain two more
explicit classes of schemes which have significantly advantages on the
subpacketization while the transmission load is equal or close to that of the
schemes constructed by Shangguan et al. (IEEE Trans. Inf. Theory, 64,
5755-5766, 2018) for the same number of users and memory size.
