# Low-Complexity Multi-Size Circular-Shift Network for 5G New Radio LDPC Decoders
## 3 Proposed Low-Complexity Multi-Size Circular-Shift Network for 5G New Radio LDPC Decoder
### Proposed Low-Complexity Multi-Size Circular-Shift Network  
The principal of our proposed low-complexity MSCN structure for 5G NR LDPC decoders is based on the fine-coarse structure and the forward routing network. The design target of the proposed architecture is focused on multi-size message-passing network with lower complexity. In general, the fine-coarse structure decomposes the system into smaller subnetworks with fixed block sizes. The block size of the subnetworks is determined by the GCD of all supported block sizes. For 5G NR LDPC codes, the block sizes vary over a very large range from 2 to 384. In this case, the MSCN structure would be made of 192 2-input pre-rotators, plus an additional stage of 2 192-input main rotators as the GCD is 2. When the size of the submatrix is 384, this approach reduces the hardware complexity compared with the direct implementation using 384 384-to-1 MUXs. However, it still suffers from significant hardware overhead. Therefore, the same technique is not directly applicable in the case of 5G NR LDPC codes. There is a need to accommodate all 51 block sizes while achieving lower complexity. To overcome the drawback, a novel MSCS network architecture based on fine-coarse structure, in which the subnetworks are reconfigurable, is presented in this section. Figure 7 illustrates the proposed low-complexity  
Figure 6: Fine-coarse structure. MSCS network structure. As shown in Figure 7b, the proposed structure can be constructed from two stages of the sub-MSCS network in which FRCS structure is conducted. First, denote the maximum block sizes of the MSCS network is \(P\), the size of the submatrix for the circular shift is \(Z\), and the value of the required circular shift is \(c\). For this fine-coarse structure, \((P_{p},Z_{p},c_{p})\) and \((P_{m},Z_{m},c_{m})\) which are the set of maximum block size, the size of the submatrix for the circular shift, and the circular-shift value for the sub-MSCS network in the pre-rotator and main rotator stage, respectively, are further defined. Based on the characteristics of the modified lifting sizes \(Z\) table in Figure 4b, the pre rotator block can be constructed from 32 FRCSs, named as \(Pre[i]\), in which the first 16 FRCSs (for \(0\leq i<16\)) are with input size \(Z_{p}=15\) (FRCS15) and the remaining 16 FRCSs for \(16\leq i<32\) are with input size \(Z_{p}=12\) (FRCS12). Similarly, the main rotator is decomposed into 15 FRCSs, denoted as \(Main[j]\). The first to the 12 th FRCS (FRCS32) (for \(0\leq j<12\)) have the same number of \(Z_{m}=32\) input blocks while the remaining 3 FRCSs (FRCS16) only comprise of \(Z_{m}=16\) input blocks for \(12\leq j<15\). In this structure, all the  
Figure 7: Proposed MSCS network structure for 5G NR LDPC decoders: (**a**) MSCS controller; (**b**) Two stages of the sub-MSCS network. control signals can be implemented with small area and determined on-the-fly. The control signal for the proposed MSCS network would be calculated from the values of \(P\), \(c\) and \(a\), as illustrated in Figure 7b. Please note that the values of \(a\) in the modified table in Figure 4b are handled instead of the original ones. The MSCS network controller consists of two parts: a circular-shift sizes generator and a circular-shift amount generator. The values of the circular shift sizes for pre rotator and main rotator subnetworks are defined as follows:  
\[ \begin{split} Z_{p}&=a,\\ Z_{m}&=\frac{Z}{a}.\end{split}  \tag{3}\]  
In the fine-coarse structure, the circular-shift amount \(c\) can be described by the sum of two terms as \(c=c_{p}+c_{m}\times Z_{p}\), where \(0\leq c_{p}<Z_{p}\) as mentioned in previous section. Therefore, the circular-shift values for the two rotators can be determined as follows:  
\[ \begin{split} c_{p}[i]=c\pmod{Z_{p}},\quad 0\leq i<32, \end{split}  \tag{4}\]  
\[ \begin{split} c_{m}[j]=\begin{cases}\frac{c}{Z_{p}}&\text{if }0 \leq j<Z_{p}-c_{p},\\ \frac{c}{Z_{p}}+1&\text{if }j\geq Z_{p}-c_{p}.\end{cases}\end{split}