## 3 TT-NOMS and TT-NSMS Channel Decoding Algorithm  
The NMS algorithm adds a multiplicative factor to the MS algorithm, and then expands it into an NNMS network decoding algorithm through the unfold method. Different from the NMS algorithm, the OMS algorithm adds an additive factor to the MS algorithm, and there is no multiplicative factor. Whether the TT and related tensor decomposition algorithms are suitable for the NOMS algorithm needs further verification. Parameters matrices are decomposed by TT in NOMS algorithm, called TT-NOMS. In Equation (7), the scalar \(\beta\) in the traditional OMS algorithm becomes the tensor \(\beta_{v^{\prime},c}^{t}\) in each iteration for each edge of NOMS algorithm, which results in additional complexity in the calculation of information transfer for CNs layers. In Equations (8) and (9), tensors \(w_{v,c}^{t}\) and \(w_{v}^{t}\) are additionally added as training parameters in the neural decoding algorithm. First, we transform the three-dimensional parameter tensors \(\beta_{v^{\prime},c^{\prime}}^{t}\), \(w_{v,c^{\prime}}^{t}\), \(w_{v}^{t}\) into four-dimensional tensors \(\beta_{v^{\prime},c^{\prime}}^{t}\), \(w_{v,c^{\prime}}^{t}\), \(w_{v}^{t}\) respectively. Then we use TT decomposition on the resulting three four-dimensional tensors as shown below,  
\[\beta_{v^{\prime},c}^{t}=\beta_{v^{\prime},c}^{1t}\beta_{v^{\prime},c}^{2t} \beta_{v^{\prime},c}^{3t}\beta_{v^{\prime},c}^{4t}. \tag{18}\]\[\mathbf{w}_{v,c}^{t}=w_{v,c}^{1t}w_{v,c}^{2t}w_{v,c}^{3t}w_{v,c}^{4t}. \tag{19}\]  
\[\mathbf{w}_{v}^{t}=w_{v}^{1t}w_{v}^{2t}w_{v}^{3t}w_{v}^{4t}. \tag{20}\]  
Substituting Equations (18)-(20) into Equations (7)-(9), we have  
\[ \begin{split} u_{c,v}^{t}=\max(\min_{v^{\prime}\in M(c\setminus v )}|u_{v^{\prime},c^{\prime}}^{t}|-\beta_{v^{\prime}}^{1t}\beta_{v^{\prime},c}^{ 2t}\beta_{v^{\prime},c}^{3t}\beta_{v^{\prime},c^{\prime}}^{4t},0)\\ \bullet\prod_{v^{\prime}\in M(c\setminus v)}sign(u_{v^{\prime},c }^{t}),\end{split}  \tag{21}\]  
\[ \begin{split} u_{v,c}^{t}=w_{v,c}^{1t}w_{v,c}^{2t}w_{v,c}^{3t}w_{ v,c}^{4t}(l_{v}+\sum_{c^{\prime}\in N(v\setminus c)}u_{c^{\prime},p}^{t-1}), \end{split}  \tag{22}\]  
\[ \begin{split} s_{v}^{t}=w_{v}^{1t}w_{v}^{2t}w_{v}^{3t}w_{v}^{4t}(l _{v}+\sum_{c^{\prime}\in N(v)}u_{c^{\prime},p}^{t}).\end{split}  \tag{23}\]  
In the TT-NOMS algorithm, we use Equations (21)-(23) to calculate CNs layers' information, VNs layers' information, and the soft output information, respectively, instead of Equations (7)-(9) in the NOMS algorithm. The loss function is then expressed as  
\[L\big{(}s^{t}\big{)}=\frac{1}{N-k}\sum_{i=1}^{N-k}max(1-softsynd(s)_{c}^{t},0). \tag{24}\]  
Syndrome loss function [24] in TT-NOMS is used in Equation (24), where \(softsynd(s)_{c}^{t}\) denotes the syndrome, t denotes the number of iteration and \(s_{v}^{t}\) denotes the estimated message. The syndrome loss calculates errors through estimated information \(s_{v}^{t}\) without the need of labels, so NOMS is called unsupervised neural network. The syndrome can be expressed as  
\[softsynd(s)_{c}^{t}=\min_{v\in M(c)}s_{v}^{t}\prod_{v\in M(c)}sign(s_{v}^{t}) \tag{25}\]  
In TT-NSMS algorithm, Equation (12) is replaced by Equation (26), two weights \(\alpha_{v^{\prime},c^{\prime}}^{t}\) and messages \(\min_{v^{\prime}\in M(c\setminus v)}|u_{v^{\prime},c^{\prime}}^{t}|\) are decomposed by TT into four low-rank weights after reshaping in high order, where \(\alpha_{v^{\prime},c}^{t}=\alpha_{v^{\prime},c}^{1t}\alpha_{v^{\prime},c}^{2t }\alpha_{v^{\prime},c^{\prime}}^{3t}\alpha_{v^{\prime},c^{\prime}}^{4t}\), \(\beta_{v^{\prime},c}^{t}=\beta_{v^{\prime},c}^{1t}\beta_{v^{\prime},c}^{2t }\beta_{v^{\prime},c}^{3t}\beta_{v^{\prime},c^{\prime}}^{4t}\), \(\min_{v^{\prime}\in M(c\setminus v)}|u_{v^{\prime},c}^{t}|=U_{v^{\prime},c}^{1t }U_{v^{\prime},c}^{2t}U_{v^{\prime},c}^{3t}U_{v^{\prime},c}^{4t}\). That is,  
\[