  Coded caching technique is an efficient approach to reduce the transmission
load in networks and has been studied in heterogeneous network settings in
recent years. In this paper, we consider a new widespread caching system called
$(K_1,K_2,U,r,M,N)$ two-dimensional (2D) caching-aided ultra-dense network
(UDN) with a server containing $N$ files, $K_1K_2$ cache nodes arranged neatly
on a grid with $K_1$ rows and $K_2$ columns, and $U$ cache-less users randomly
distributed around cache nodes. Each cache node can cache at most $M\leq N$
files and has a certain service region by Euclidean distance. The server
connects to users through an error-free shared link and the users in the
service region of a cache node can freely retrieve all cached contents of this
cache node. We aim to design a coded caching scheme for 2D caching-aided UDN
systems to reduce the transmission load in the worst case while meeting all
possible users' demands. First, we divide all possible users into four classes
according to their geographical locations. Then our first order optimal scheme
is proposed based on the Maddah-Ali and Niesen scheme. Furthermore, by
compressing the transmitted signals of our first scheme based on Maximum
Distance Separable (MDS) code, we obtain an improved order optimal scheme with
a smaller transmission load.
