Hierarchical Coded Caching in High Memory Regime with Coded Placement

We consider a two-layer hierarchical coded caching network where a server with a library of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$N$</tex> files is connected to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$K_{1}$</tex> mirrors, each having a cache memory of size <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$M_{1}$</tex> files. Each mirror is further connected to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$K_{2}$</tex> users, each equipped with a dedicated cache of size <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$M_{2}$</tex> files. In this paper, we propose two distinct coded caching schemes based on coded placement, corresponding to two distinct memory pairs, (<tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$M_{1}, M_{2}$</tex>), which operate effectively in the high memory regime, i.e., when both <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$M_{1}$</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$M_{2}$</tex> are large. We show that the proposed schemes outperform existing schemes at these memory points for smaller values of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$K_{2}$</tex>. In setups where mirrors are positioned near each other, avoiding signal interference is crucial. This can be ensured by having all mirrors transmit using orthogonal carrier frequencies. To compare our schemes with existing ones, we used the composite rate metric, which accurately represents the total bandwidth utilized in such setups. The composite rate is given by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\bar{R}=R_{1}+K_{1} R_{2}$</tex>, where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$R_{1}$</tex> is the rate from the server to the mirrors, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$R_{2}$</tex> is the rate from the mirrors to the users, with respect to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$M_{1}$</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$M_{2}$</tex>.