  Coded caching scheme, which is an effective technique to increase the
transmission efficiency during peak traffic times, has recently become quite
popular among the coding community. Generally rate can be measured to the
transmission in the peak traffic times, i.e., this efficiency increases with
the decreasing of rate. In order to implement a coded caching scheme, each file
in the library must be split in a certain number of packets. And this number
directly reflects the complexity of a coded caching scheme, i.e., the
complexity increases with the increasing of the packet number. However there
exists a tradeoff between the rate and packet number. So it is meaningful to
characterize this tradeoff and design the related Pareto-optimal coded caching
schemes with respect to both parameters.
  Recently, a new concept called placement delivery array (PDA) was proposed to
characterize the coded caching scheme. However as far as we know no one has yet
proved that one of the previously known PDAs is Pareto-optimal. In this paper,
we first derive two lower bounds on the rate under the framework of PDA.
Consequently, the PDA proposed by Maddah-Ali and Niesen is Pareto-optimal, and
a tradeoff between rate and packet number is obtained for some parameters.
Then, from the above observations and the view point of combinatorial design,
two new classes of Pareto-optimal PDAs are obtained. Based on these PDAs, the
schemes with low rate and packet number are obtained. Finally the performance
of some previously known PDAs are estimated by comparing with these two classes
of schemes.
