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In caching system, it is desirable to design a coded caching scheme with thetransmission load $R$ and subpacketization $F$ as small as possible, in orderto improve efficiency of transmission in the peak traffic times and to decreaseimplementation complexity. Yan et al. reformulated the centralized codedcaching scheme as designing a corresponding $F\times K$ array called placementdelivery array (PDA), where $F$ is the subpacketization and $K$ is the numberof users. Motivated by several constructions of PDAs, we introduce a frameworkfor constructing PDAs, where each row is indexed by a row vector of some matrixcalled row index matrix and each column's index is labelled by an element of adirect product set. Using this framework, a new scheme is obtained, which canbe regarded as a generalization of some previously known schemes. When $K$ isequal to ${m\choose t}q^t$ for positive integers $m$, $t$ with $t<m$ and $q\geq2$, we show that the row index matrix must be an orthogonal array if all theusers have the same memory size. Furthermore, the row index matrix must be acovering array if the coded gain is ${m\choose t}$, which is the maximal codedgain under our framework. Consequently the lower bounds on the transmissionload and subpacketization of the schemes are derived under our framework.Finally, using orthogonal arrays as the row index matrix, we obtain two moreexplicit classes of schemes which have significantly advantages on thesubpacketization while the transmission load is equal or close to that of theschemes constructed by Shangguan et al. (IEEE Trans. Inf. Theory, 64,5755-5766, 2018) for the same number of users and memory size.

A framework of constructing placement delivery arrays for centralized coded caching