# LDPC Decoder Design Using Compensation Scheme of Group Comparison for 5G Communication Systems
## 3 LDPC Decoding Algorithms
### NPMSa  
Generally, the critical path of the decoder hardware design employing the NMSA is the operation of the check node process. This is because numerous comparator operations cause severe operational delays. Therefore, the NPMSA was proposed to speed up the overall operation and relieve the hardware burden of the NMSA, which includes area and routing costs. The NPMSA adopts a grouping search that divides the prior message \(Q^{k}_{i,j}\) into G groups, to replace the sorter used by the NMSA. In each group of the NPMSA, the architecture only needs to be implemented using 2-to-1 comparators [20]. A significant reduction in the architecture overhead is noted when compared to the sorter using 2-to-2 comparators. The gap of the hardware burden between the grouping search and sorter increases as the number of prior messages \(Q^{k}_{i,j}\) increases. However, in alleviating the effects of hardware burden, a grouping search cannot accurately determine the correct second minimum. This is because the correct second minimum is typically lost after every 2-to-1 comparator operation in the hardware design. This causes the accuracy of the algorithm to decrease and directly affects the decoding capability. The formula for updating the check nodes with a grouping search is expressed as:  
\[R^{k}_{i,j}\ =\ \gamma\cdot\left(\prod_{j^{\prime}\in N(c)\backslash j}sgn \left(Q^{k}_{i,j}\right)\right)\left\{\begin{array}{l}min^{1 st},\qquad j^{ \prime}\
eq j_{index}\\ min^{2 nd},\qquad j^{\prime}\ =\ j_{index}\end{array}\right., \tag{5}\]  
where:  
\[min^{1 st}\ =\ \underset{j\in N(c)}{min}\left|Q^{k}_{i,j}\right| \tag{6}\]and the probabilistic second minimum is:  
\[min^{2 nd}\ =\ max\left\{\min_{j\in G_{L}}\left|Q_{i,j}^{k}\right|,\min_{j\in G_{R}} \left|Q_{i,j}^{k}\right|\right\}. \tag{7}\]  
The index \(j_{index}\) denotes the position of the first minimum. \(G_{L}\) and \(G_{R}\) represent the left and the right groups of the prior message \(Q_{i,j}^{k}\), respectively, when there are two groups.  
Table 1 lists a comparison of the check node process executed using the NMSA, the split-row threshold algorithm (SRTA) [21], and NPMSA. The two-input comparison was used as the basis for comparison. From an algorithmic perspective, the NPMSA can reduce the number of two-input comparisons by 49.1%, with a 0.05 dB loss in the error-rate performance, compared to the NMSA. Compared to the SRTA, the NPMSA can reduce the number of two-input comparisons by 35.4%, with a 0.15 dB enhancement in error-rate performance.  
It is noteworthy that the probabilistic second minimum value in Equation (7) is equal to the correct second minimum value in Equation (3) when the correct second minimum value is not in the same group of the correct first minimum value. This means that the correct second minimum value can be easily achieved when the number of groups becomes larger. However, a large number of groups intensifies the hardware burden because of the increment of 2-to-2 comparators. Additionally, the occurrence of differences between the correct and probabilistic second minimum values results in a decline in the decoding capability. The NPMSA suffers from a high error floor region, which was revealed in [18]. Thus, the difference between the correct and probabilistic second minimum values must be reduced when the difference occurs.