## 4 Numerical Results  
In simulations, we use the conventional 16-QAM constellation and apply the look-up table algorithm in [14; 15] to differentially encode the information. Particularly, one 4-bit information symbol is encoded into two consecutive 16-QAM points; thus, the non-coherent distance as in Equation (15) calculated with the length of transmitted signal vectors \(\mathbf{x}_{1},\mathbf{x}_{2}\) is 2. Finally, the look-up table for 16-DQAM has 16 rows presenting 16 different groups; each group contains 16 different vectors \(\mathbf{x}\), and all transmitted signal vectors in the same group correspond to the same information symbol. The look-up table for the proposed 16-DQAM scheme with Rician fading channel \(K_{r}=1\) and an average \(\text{SNR}=-4\) dB is given in Table 1 as an example. Notice that, in [14; 15] after generating the look-up table, there is one more step that maps information bit to each group. This step adds a little more improvement in bit error rate performance since groups with small non-coherent distance are mapped to information bit symbols with a small difference in the number of bits. However, in this simulation, we focus on comparing the performance of detector and distance between our proposed ones and previous ones in [12; 13] so that we skip this step and add the information bit symbols sequentially from first group to last group in the look-up table. Figure 2 illustrates the simulation results of the proposed 16-DQAM scheme with different lengths of estimated signal vector \(L=2\) and \(3\), as well as the 16-DQAM schemes in [12; 13] where the number of BS antennas are \(M=128\) and \(500\), and the coherent length \(T=7\). Since the authors of [12] did not show the detector when \(M\) is finite, we suppose that the 16-DQAM scheme in [12] uses the generalized detector as in [13], Equation (6), and the corresponding non-coherent distance in [13]. As previously shown in [12; 13], the 16-DQAM scheme of [12] shows an error floor when \(M\) is not very large. The proposed scheme significantly outperforms the schemes in [12; 13] for both \(M=128\) and \(500\) \begin{table}
\begin{tabular}{l l l l l} \hline \hline \(G_{1}\) & \((0;0)\) & \((1;1)\) & \((2;2)\) & \((3;0)\) \\ \(G_{2}\) & \((0;1)\) & \((1;6)\) & \((2;0)\) & \((3;1)\) \\ \(G_{3}\) & \((0;2)\) & \((1;0)\) & \((2;13)\) & \((3;2)\) \\ \(G_{4}\) & \((0;3)\) & \((1;3)\) & \((2;3)\) & \((3;3)\) \\ \(G_{5}\) & \((0;4)\) & \((1;5)\) & \((2;6)\) & \((3;4)\) \\ \(G_{6}\) & \((0;5)\) & \((1;10)\) & \((2;4)\) & \((3;5)\) \\ \(G_{7}\) & \((0;6)\) & \((1;4)\) & \((2;1)\) & \((3;6)\) \\ \(G_{8}\) & \((0;7)\) & \((1;7)\) & \((2;7)\) & \((3;7)\) \\ \(G_{9}\) & \((0;8)\) & \((1;9)\) & \((2;10)\) & \((3;8)\) \\ \(G_{10}\) & \((0;9)\) & \((1;14)\) & \((2;8)\) & \((3;9)\) \\ \(G_{11}\) & \((0;10)\) & \((1;8)\) & \((2;5)\) & \((3;10)\) \\ \(G_{12}\) & \((0;11)\) & \((1;11)\) & \((2;11)\) & \((3;11)\) \\ \(G_{13}\) & \((0;12)\) & \((1;13)\) & \((2;14)\) & \((3;12)\) \\ \(G_{14}\) & \((0;13)\) & \((1;2)\) & \((2;12)\) & \((3;13)\) \\ \(G_{15}\) & \((0;14)\) & \((1;12)\) & \((2;9)\) & \((3;14)\) \\ \(G_{16}\) & \((0;15)\) & \((1;15)\) & \((2;15)\) & \((3;15)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Look-up table for 16-DQAM using proposed distance, \(L=2\). Figure 2: Performance comparison between the proposed 16-DQAM scheme and the previous works of References [12; 13] under Rayleigh fading. Since the schemes in [12; 13] can only detect two consecutive symbols at a time, we simulate our proposed scheme when the length of signal vector is \(L=2\). We can clearly see that, with the same channel condition and signal vector's length, the proposed scheme outperforms the other schemes for \(M=128\) with nearly 3 dB when BER = \(10^{-4}\). With not so large number of BS antenna \(M=128\), the significant improvement of the proposed scheme shows a huge potential that it can be deployed in real system.