  In this paper, we focus on the hopping pattern design for device-to-device
(D2D) discovery. The requirements of hopping pattern is discussed, where the
impact of specific system constraints, e.g., frequency hopping, is also taken
into consideration. Specifically speaking, we discover and utilize the novel
feature of resource hopping, i.e., "hopping invariant" to design four new
hopping patterns and analyze their performance. The hopping invariant can be
used to deliver information for specific users without extra radio resources,
and due to the connection between hopping invariant and resource location,
receiver complexity can be significantly reduced. Furthermore, our schemes are
designed to be independent of discovery frame number, which makes them more
suitable to be implemented in practical systems.
