  Differential spatial modulation (DSM) was recently proposed to overcome the
challenge of channel estimation in spatial modulation (SM). In this letter, we
propose a gray code order of antenna index permutations for DSM. To facilitate
the implementation, the well-known Trotter-Johnson ranking and unranking
algorithms are adopted, which results in similar computational complexity to
the existing DSM that uses the lexicographic order. The coding gain achieved by
the proposed gray code order over the existing lexicographic order is also
analyzed and verified via simulations, which reveals a maximum of about 1.2dB
for the case of four transmit antennas. Based on the gray coding framework, we
further propose a diversity-enhancing scheme named intersected gray (I-gray)
code order for DSM, where the permutations of active antenna indices are
selected directly from the odd (even) positions of the full permutations in the
gray code order. From analysis and simulations, it is shown that the I-gray
code order can harvests an additional diversity order at the expense of only
one information bit loss for each transmission with respect to the gray code
order.
