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Caching is a commonly used technique in content-delivery networks which aimsto deliver information from hosting servers to users in the most efficient way.In 2014, Maddah-Ali and Niessen formulated caching into a formal informationtheoretic problem and it has gained a lot of attention since then. It is knownthat the caching schemes proposed by Ali-Niesen and Yu et. al. are optimal,that is, they require the least number of transmissions from the server tosatisfy all users' demands. However for these schemes to work, each file needsto be partitioned into $F^*$ subfiles ($F^*$ is called the subpacketizationlevel of files) with $F^*$ growing exponentially in the number $K$ of users. Asa result, it is problematic to apply these schemes in practical situations,where $K$ tends to be very large. There rise the following questions: (1) arethere optimal schemes in which each file is partitioned into $F$ subfiles,where $F$ is not exponential, say polynomial for example, in $K$? (2) if theanswer to this question is no, is there a near-optimal scheme, a scheme whichis as asymptotically good as the one in \cite{ali1,yu}, with $F$ polynomial in$K$? Both these questions are open.  Our main contribution in this paper is to provide answers to above questions.Firstly, we prove that under some mild restriction on user's cache rate, thereare no optimal schemes with $F$ smaller than $F^*$. Moreover, we give necessaryand sufficient conditions for the existence of optimal schemes in this case.Secondly, we provide an affirmative answer to the second question raised aboveby an explicit construction and a detailed performance analysis.

Symmetric uncoded caching schemes with low subpacketization levels