# Strategy for Non-Orthogonal Multiple Access and Performance in 5G and 6G Networks
## 5 MA with Code Separation
### Grassmannian-Sequence-Based NOMA Protocols Founded on the Generalized Welch Equality  
Grassmannian sequences provide the foundation for LG Electronics' NCMA proposal, which makes use of these sequences. When creating these kinds of sequences, a more severe optimization strategy is applied in comparison to the WSMA scheme. This strategy involves minimizing the greatest level of mutual correlation that can exist between any two WBE sequences. A linear space-packing problem characterizes the process of generating Grassmannian sequences. It is possible to define the set of Grassmannian sequences as follows:  
\[\sum_{l=1}^{K}\sum_{l=1}^{K}P_{l}P_{l}\big{|}S_{i}S_{j}\big{|}^{2}\geq\Big{(} \sum_{k=1}^{K}P_{k}\Big{)}^{2}/N. \tag{5}\]  
\[R_{s}=\big{\|}S^{\prime}PS\big{\|}^{2}=\sum_{i=1}^{k}\sum_{j=1}^{k}P_{i}P_{j} \big{|}S_{i}^{\prime}S_{j}\big{|}^{2} \tag{6}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **No. of Sequences** & **1** & **2** & **3** & **4** \\ \hline \multirow{4}{*}{Sequence count down} & 1 & \(-0.661+0.101\)i & \(-0.122+0.521\)i & \(0.526-0.428\)i & \(0.291-0.523\)i \\ \cline{2-5}  & 2 & \(0.096+0.582\)i & \(-0.432-0.095\)i & \(0.254-0.323\)i & \(0.621+0.254\)i \\ \cline{2-5}  & 3 & \(-0.432-0.291\)i & \(-0.583+0.382\)i & \(-0.391-0.167\)i & \(-0.591-0.342\)i \\ \cline{2-5}  & 4 & \(-0.218+0.362\)i & \(0.482+0.241\)i & \(0.620-0.091\)i & \(-0.494+0.014\)i \\ \hline \multirow{4}{*}{Sequence count down} & **No. of Sequences** & **5** & **6** & **7** & **8** \\ \cline{2-5}  & 1 & \(-0.572-0.214\)i & \(-0.341+0.125\)i & \(0.421+0.118\)i & \(0.470+0.213\)i \\ \cline{2-5}  & 2 & \(0.061-0.644\)i & \(0.367-0.143\)i & \(-0.024-0.561\)i & \(0.048-0.424\)i \\ \cline{2-5}  & 3 & \(0.353-0.288\)i & \(0.651-0.066\)i & \(-0.450+0.098\)i & \(0.407+0.160\)i \\ \cline{2-5}  & 4 & \(-0.107-0.461\)i & \(0.217+0.486\)i & \(-0.516+0.111\)i & \(-0.490+0.362\)i \\ \hline \hline \end{tabular}
\end{table} 
Table 1: A set \(S\) of WBE sequences from the 3 GPP report. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **TSC** & \(\boldsymbol{\mu}\) & \(\boldsymbol{d_{cord}}\) \\ \hline \(\begin{bmatrix}1&p_{12}&p_{13}&p_{14}\\ p_{12}&1&p_{23}&p_{24}\\ p_{13}&p_{23}&1&p_{34}\\ p_{14}&p_{24}&p_{34}&1\end{bmatrix}\) & \(\begin{bmatrix}1&p&p&p\\ p&1&p&p\\ p&p&p&1\end{bmatrix}\) & \(\begin{bmatrix}1&0&p_{13}&p_{14}\\ 0&1&p_{23}&p_{24}\\ p_{13}&p_{23}&1&0\\ p_{14}&p_{24}&0&1\end{bmatrix}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Optimizations of various parameters, and relative impact on the Gram determinant. The matrix is \(G=[S_{1}\ldots S_{K}]\), which consists of vectors of sequential values and \(G\subset C^{N\times K}\) where \(C^{N\times K}\) is the set of complex matrices with the dimensions \(N\times K\), \(N\) is the length of the sequences, and \(K\) is the total number of sequences. The NTT DOCOMO Company's UGMA system utilizes generalized Welch-bound equality (GWBE) sequences in order to division subscriber signals [30]. The GWBE is the foundation for these sequences, in contrast to the WBE sequences that correspond to the Welch bound (2). To the Belch border, this encompasses the following components in the total power \(P_{k}\) of signals \(K\) of subscribers:  
The GWBE sequences that were just described are made up of complex numbers, the real and imaginary components can be quantized utilizing discrete value ranges. ZTE's MUSA method [31] likewise makes use of sequences that contain elements that have been quantized. The QPSK and QAM signal constellations can provide the building blocks for the code sequences, which can then be used. For instance, the number of such sequences is equal to times the length of the sequences \(4^{N}\) in the case of QPSK, where \(N\) is the length of the sequences. The 3 GPP document has other examples of sequences. These sequences are created using a computer optimization process to establish a low mutual correlation between sequences with varying cyclic shifts.