  We consider a coded caching network consisting of a server with a library of
$N$ files connected to $K$ users, where each user is equipped with a dedicated
cache of size $M_p$ units. In addition to that, the network consists of
$\Lambda\leq K$ helper caches, each with a size $M_h$ units. Each helper cache
can serve an arbitrary number of users; however, each user can access only a
single helper cache. Also, we assume that the server knows the user-to-helper
cache association, defined as the sets of users connected to each helper cache,
during the cache placement phase. We propose a solution for the aforementioned
coded caching problem by introducing a combinatorial structure called a Shared
and Private Placement Delivery Array (SP-PDA). These SP-PDAs describe the
helper cache placement, private cache placement, and the server transmissions
in a single array. Further, we propose a novel construction of SP-PDAs using
two Placement Delivery Arrays (PDAs). Interestingly, we observe that the
permutations of the columns of the two chosen PDAs result in SP-PDAs with
different performances. Moreover, we characterize the conditions for selecting
the best column permutations of the chosen PDAs. Furthermore, the coded caching
schemes resulting from SP-PDAs subsume two existing coded caching schemes as
special cases. Additionally, SP-PDAs enable the construction of coded caching
schemes with much smaller subpacketization numbers -subpacketization number is
defined as the number of subfiles to which a file is divided- compared to the
existing schemes, without paying much in terms of rate (the size of the
transmission in the delivery phase).
