We consider a cache-aided interference network which consists of a library of N files, K T transmitters and K R receivers (users), each equipped with a local cache of size M T and M R files respectively, and connected via a discrete-time additive white Gaussian noise channel. Each receiver requests an arbitrary file from the library. The objective is to design a cache placement without knowing the receivers' requests and a communication scheme such that the sum Degrees of Freedom (sum-DoF) of the delivery is maximized. This network model has been investigated by Naderializadeh et al., who proposed a prefetching and a delivery scheme that achieve a sum-DoF of min{M T K T + K R M R /N, K R }. One of the biggest limitations of this scheme is the requirement of high subpacketization level. This paper attempts to design new algorithms to reduce the file subpacketization in such a network. In particular, we propose a new approach for both prefetching and linear delivery based on a combinatorial design called hypercube. We show that the required number of packets per file can be exponentially reduced compared to the state-of-the-art scheme proposed by Naderializadeh et al., or the NMA scheme. When M T K T + K R M R ≤ K R , the achievable one-shot sum-DoF using this approach is M T K T + K R M R /N, which shows that 1) the one-shot sum-DoF scales linearly with the aggregate cache size in the network and 2) it is within a factor of 2 to the information-theoretic optimum. Surprisingly, the identical and near optimal sum-DoF performance can be achieved using the hypercube approach with a much less file subpacketization.