A Novel Construction of Coded Caching Schemes with Polynomial Subpacketizations via Projective Geometry

Ina <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(K, M, N)$</tex> coded caching scheme, in order to pursue a low broadcasting rate based on the designed placements at each user's cache we have to divide each file stored in the server into certain packets. However, the implementation complexity of this scheme increases with the number of packets. So it is important to design a scheme with a small subpacketization level and a relatively low transmission load. Placement delivery array (PDA) is a powerful combinatorial structure to characterize the coded caching scheme including the subpacketization and transmission load under uncoded placement. In this paper, using the projective geometry PG <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(q,n)$</tex> for any prime power <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$q$</tex> and positive integer <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n \geq 3$</tex>, we propose a novel class of coded caching scheme with polynomial subpacketization which is less than <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$K^{\lceil\frac{n+2}{t}\rceil-1}$</tex>. In addition, when <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$t=2$</tex>, our scheme has <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$K=\frac{q^{n}-1}{q-1}, M/N &lt; \frac{2}{q}$</tex> and subpacketization <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$F &lt; K^{2}$</tex>. Compared to the - optimal schemes under uncoded placement which have the subpacketization exponentially increasing with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$K$</tex>, our load increases at most <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\frac{K}{2 q\binom{n+1}{2}}$</tex> times; compared to the existing schemes with low subpacketizations our schemes have also advantages on the subpacketization and load.