Device-to-Device (D2D) assisted coded caching is a promising approach to improve the communication efficiency over networks. However, the basic D2D coded caching scheme requires a subpacketization size that increases exponentially with the number of users. This is infeasible since the file size needs to be extremely large in the server. It is desirable to design a scheme that achieves a small subpacketization size while keeping the rate low. Recently, D2D placement delivery array (DPDA) was proposed to address the high subpacketization issue of D2D coded caching. This paper investigates the design of DPDA from the perspectives of linear algebraic and additive combinatorics. It is shown that a linear subspace possessing certain property can be employed in the design of DPDA. Based on this, a new D2D coded caching scheme with a subquadratic subpacketization size is derived through shortening the binary Reed-Muller codes. In order to obtain a D2D coded caching scheme with a linear subpacketization size, a new combinatorial structure called proper disjoint 3-term arithmetic progression (3-AP) free set is further introduced, and a deterministic algorithm for constructing it is provided with a polynomial complexity. Both the theoretical and numerical results reveal that the proposed schemes have a superior performance in terms of subpacketization size or transmission rate.