## 2 Convert Information Mapping Aided GSDM
### Proposed CIM-GSDM Scheme  
For the sake of further improvement, the GSDM system is able to combine with CIM in which covert information is mapped with a deliberate design in the GSDM system. In the proposed CIM-GSDM scheme, the non-covert information and the covert information are considered together at Alice. The proposed scheme are still based on the GSDM system, and the difference and novelty lie in that the information delivered by receiver subsets are not only designed by indices, but also the covert information added in. Since the covert information is related to the past message, [PERSON] can demodulate correctly owing to knowing the rule while [PERSON] falls into confusion from the beginning. Therefore, it is distinct to the GSDM system and improve the security of transmission to a next level in the presence of CIM. Under this scenario, the data bits to send at a time are separated into three parts, as \(k=k_{1}+k_{2}+k_{3}\), which are exploited to design as the covert message and APM symbols. Specifically, \(k_{1}+k_{2}\) bits are employed as selection of receiver subset indices with the aid of CIM design and remaining \(k_{3}\) bits are applied for APM symbols. Based on CIM, all receivers are divided into two groups \(\mathcal{I}_{1}\) and \(\mathcal{I}_{2}\), in which \(\mathcal{I}_{1}=\left\{1,2,\cdots,\frac{N_{r}}{2}\right\}\) and \(\mathcal{I}_{2}=\left\{\frac{N_{r}}{2}+1,\frac{N_{r}}{2}+2,\cdots,N_{r}\right\}\) include the former half receivers and the latter half receivers, respectively. In this case, there are only \(f_{2}=2^{\left\lfloor\log_{2}\mathcal{C}_{N_{r}/2}^{N_{r}}\right\rfloor}\) receiver combinations in each group at Bob with \(k_{2}=\log_{2}f_{2}\). Thus, \(2f_{2}\) groups can be modulated in total. The main idea of CIM-GSDM is to use \(k_{1}\) bits to choose the receiver group in conjunction with message at last slot and use \(k_{2}\) bits to select the receivers combination index in the group. In detail, if \(K_{1}=[1]\), the current receiver group is the same as the previous one. Otherwise, the current receiver group is different from the previous one if \(K_{1}=[0]\), that is to say, choosing another receiver group. Supposing that the receiver group index selected to modulate at the \((k-1)\)-th slot is \(\mathcal{I}_{1}\), the receiver group index \(\mathcal{I}_{gk}\) chosen at the \(k\)-th slot can be represented as  
\[\mathcal{I}_{gk}=\left\{ \begin{array}{ll}\mathcal{I}_{2},&\text{ if }K_{1}=[0]\text{ and }\mathcal{I}_{gk-1}=\mathcal{I}_{1}\\ \mathcal{I}_{1},&\text{ if }K_{1}=[1]\text{ and }\mathcal{I}_{gk-1}=\mathcal{I}_{1}. \end{array} \right. \tag{9}\]  
Analogously, if the receiver group index selected to modulate at \((k-1)\)-th slot is \(\mathcal{I}_{2}\), then \(\mathcal{I}_{gk}\) can be represented as  
\[\mathcal{I}_{gk}=\left\{ \begin{array}{ll}\mathcal{I}_{1},&\text{ if }K_{1}=[0]\text{ and }\mathcal{I}_{gk-1}=\mathcal{I}_{2}\\ \mathcal{I}_{2},&\text{ if }K_{1}=[1]\text{ and }\mathcal{I}_{gk-1}=\mathcal{I}_{2}. \end{array} \right. \tag{10}\]  
For instance, an example with \(N_{r}=4\) for the proposed CIM-GSDM scheme is shown in Figure 2. In a nutshell, the design of selection for receiver groups at Bob takes the message of the former and the latter into consideration together and provide further security. After the receiver group is fixed, \(k_{2}\) bits are employed to determine the receive subset index in the certain group \(\mathcal{I}_{gk}\) as \(\mathcal{I}_{gk}(p)\),\((p=1,2,\ldots,f_{2})\). Then the symbol vector produced by [PERSON] can be expressed as  
\[\mathbf{x}_{\mathcal{I}_{gk}}^{s}=\mathbf{I}_{\mathcal{I}_{gk}}\mathbf{b}_{s}, \tag{11}\]  
where \(\mathbf{I}_{\mathcal{I}_{gk}}\) consists of \(N_{u}\) column vector chosen from identity matrix \(\mathbf{I}_{N_{t}\times N_{t}}\) corresponding to \(\mathcal{I}_{gk}\), \(\mathbf{b}_{s}(s=1,2,\ldots,2^{k_{3}})\) is a column vector built up by \(N_{u}\) symbols selected from conventional \(M\)-ary APM symbols.