We consider the coded caching problem with shared caches where several users share a cache, but each user has access to only a single cache. For this network, the fundamental limits of coded caching are known for centralized and decentralized settings under uncoded placement. In the centralized case, to achieve the gains offered by coded caching, one requires a subpacketization which increases exponentially with the number of caches. The dedicated cache networks had a similar issue, and placement delivery arrays (PDAs) were introduced as a solution to it. Using the PDA framework, we propose a procedure to obtain new coded caching schemes for shared caches with lower sub-packetization requirements. The advantage of this procedure is that we can transform all the existing PDA structures into coded caching schemes for shared caches, thus resulting in low sub-packetization schemes. We also show that the optimal scheme given by Parrinello, Ünsal and Elia (Fundamental Limits of Coded Caching with Multiple Antennas, Shared Caches and Uncoded Prefetching) can be recovered using a Maddah-Ali Niesen PDA.

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This paper considers the secretive coded caching problem with shared caches in which no user must have access to the files that it did not demand. In a shared cache network, the users are served by a smaller number of helper caches and each user is connected to exactly one helper cache. To ensure the secrecy constraint in shared cache networks, each user is required to have an individual cache of at least unit file size. For this setting, a secretive coded caching scheme was proposed recently in the literature ("Secretive Coded Caching with Shared Caches", in IEEE Communications Letters, 2021), and it requires a subpacketization level which is in the exponential order of the number of helper caches. By utilizing the PDA constructions, we propose a procedure to obtain new secretive coded caching schemes for shared caches with reduced subpacketization levels. We also show that the existing secretive coded caching scheme for shared caches can be recovered using our procedure. Furthermore, we derive a lower bound on the secretive transmission rate using cut-set arguments and demonstrate the order-optimality of the proposed secretive coded caching scheme.

Decentralized coded caching scheme, introduced by Maddah-Ali and Niesen, assumes that the caches are filled with no coordination. This work identifies a decentralized coded caching scheme -under the assumption of uncoded placement-for shared cache network, where each cache serves multiple users. Each user has access to only a single cache and the number of caches is less than or equal to the number of users. For this setting, we derive the optimal worst-case delivery time for any user-tocache association profile where each such profile describes the number of users served by each cache. The optimality is shown using an index-coding based converse. Further, we improve the delivery scheme to accommodate redundant demands. Also, an optimal linear error correcting delivery scheme is proposed for the worst-case demand scenario. Next, we consider the Least Recently Sent (LRS) online coded caching scheme where the caches need to be updated based on the sequence of demands made by the users. Cache update happens if any of the demanded file was not partially cached at the users. The update is done by replacing the least recently sent file with the new file. But, the least recently sent file need not be unique. In that case, there needs to be some ordering of the files which are getting partially cached, or else centralized coordination would have to be assumed which does not exist. If each user removes any of the least recently used files at random, then the next delivery phase will not serve the purpose. A modification is suggested for the scheme by incorporating the ordering of files. The fix is suggested so that the results on the error-correction for the online coded caching scheme introduced in [N. S. Karat, K. L. V. Bhargav and B. S. Rajan, "On the Optimality of Two Decentralized Coded Caching Schemes With and Without Error Correction," IEEE ISIT, Jun. 2020] are not affected. Moreover, all the above results with shared caches are extended to the online setting.

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