The Cycle-Concentrating PEG Algorithm for Protograph Generalized LDPC Codes

In this paper, we propose the cycle-concentrating progressive edge growth (CC-PEG) algorithm for lifting protograph generalized low-density parity-check (GLDPC) codes. In GLDPC codes, undoped variable nodes (VNs) that are not connected to generalized constraint (GC) nodes are more vulnerable to channel errors than doped VNs protected by GC nodes. We observe that among GLDPC codes sharing the same protograph structure, codes with fewer local cycles at undoped VNs have better decoding performances. Inspired by this observation, the CC-PEG algorithm is proposed to concentrate local cycles at doped VNs and avoid local cycles at vulnerable undoped VNs during the lifting process. Specifically, the CC-PEG algorithm first collects edges that result in the maximum undoped girth, defined as the length of the shortest cycle containing undoped VNs. Following this, the CC-PEG algorithm selects the edge with the lowest concentrated cycle metric. Consequently, the lifted codes exhibit asymmetric cycle distributions concentrated around robust doped VNs. Simulation results for various protographs show that the CC-PEG algorithm achieves a performance gain of up to 20 times lower frame error rate compared to conventional lifting algorithms.


I. INTRODUCTION
Generalized low-density parity-check (GLDPC) codes [1] are generalized versions of LDPC codes. Theoretically, GLDPC codes have a larger minimum distance [2], faster decoding convergence [3], and better asymptotic performance than LDPC codes [4]. Moreover, practically constructed GLDPC codes have outperformed various codes for low-latency communication [5], which suggests that GLDPC codes can be a promising class of codes for the next generation communication systems. GLDPC codes incorporate not only single parity check (SPC) nodes but also generalized constraint (GC) nodes as their check nodes (CNs). GC nodes The associate editor coordinating the review of this manuscript and approving it for publication was Luca Barletta. represent arbitrary linear component codes [6] and provide stronger protection to their neighbor variable nodes (VNs). One way to construct GLDPC codes is doping [7], replacing some SPC nodes in LDPC codes with GC nodes. We call VNs connected to GC nodes the doped VNs. In contrast, the undoped VNs are only connected with SPC nodes and thus more vulnerable to channel errors than the doped VNs.
The progressive edge growth (PEG) algorithm [8] is the most popular algorithm for lifting protograph-based LDPC codes [9]. In an edge-by-edge manner, the PEG algorithm adds an edge to the current Tanner graph while maximizing the local girth of the target VN. Due to this greedy behavior, local cycles tend to be more generated in the latter lifted VNs. The PEG algorithm first arranges VNs according to their degrees in a non-decreasing order. As a result, VNs with lower degrees tend to have fewer local cycles than VNs with higher degrees, which makes the lifted graph have a better overall approximate cycle extrinsic message degree (ACE) characteristic. While some studies have employed the PEG algorithm for constructing certain GLDPC codes [10], [11], no specialized lifting method has been developed for an arbitrarily given GLDPC protograph. Also, while some PEG-based algorithms using girth [12] and cycle metrics (CMs) [13] have optimized LDPC codes to get a better error floor performance, no PEG method has yet been proposed that employs CMs to establish a cycle distribution suited to GLDPC codes.
In this paper, following our observation in [14], we first demonstrate that the lifting order of VNs in the PEG algorithm affects the cycle distribution and the decoding performance of GLDPC codes. The GLDPC codes constructed by lifting undoped VNs first have fewer local cycles at undoped VNs and better performance than GLDPC codes with different lifting orders. Based on this result, we propose a VN lifting order and a new lifting algorithm called cycle-concentrating PEG (CC-PEG) specialized for GLDPC codes. We first sort the lifting order of VNs according to their degrees and doping status. Then, the CC-PEG evaluates the cycle distribution of the current Tanner graph by calculating VN CMs M v s using the parallel vector message passing (PMP) algorithm [13]. Lastly, the CC-PEG connects edges in a way that generates fewer local cycles at vulnerable undoped VNs. While this connection method results in the concentrated local cycles at doped VNs, the doped VNs can maintain the reliability through the connected GC nodes. Fig. 1 illustrates a protograph for a GLDPC code and the core concept of the CC-PEG algorithm.
Extensive decoding simulations are conducted over both the binary erasure channel (BEC) and the additive white Gaussian noise (AWGN) channel using regular and irregular GLDPC protographs in [15] and [16], which have capacity-approaching performance. The results confirm that the CC-PEG algorithm achieves a significant coding gain compared to other existing lifting algorithms [9], [13], particularly for large protographs and moderate code lengths, where GLDPC codes are known to perform better than LDPC codes [5].
The contributions of this paper can be summarized as follows. First, we propose the CC-PEG algorithm which is the first lifting algorithm specially designed for GLDPC codes. Although many studies have focused on the optimization of protograph GLDPC codes [15], [16], [17], a lifting method specialized for GLDPC codes has yet to be developed. The proposed algorithm can be applied to lift any GLDPC protograph with doped and undoped VNs. Second, to the best of our knowledge, this is the first work that intentionally creates more cycles at specific nodes to improve the decoding performance. The concept of the CC-PEG algorithm can further be applied to lift LDPC codes for non-uniform channels [18], where partial protection is required. Third, we propose new cycle and girth metrics to evaluate lifted GLDPC Tanner graphs. As the theoretical analysis of GLDPC codes [19] relies on the protograph structure, it is challenging to analyze the effect of the different lifting methods because the lifted codes have the same protograph strucuture. Therefore, we quantitatively analyze the reasons for the performance gains achieved by CC-PEG codes (GLDPC codes lifted from a given protograph using the CC-PEG algorithm) based on the proposed metrics related to GLDPC graph structure. Further, the proposed metrics can also be used to evaluate general GLDPC codes regardless of the protograph construction.
The rest of this paper is organized as follows. Section II introduces the notations and preliminaries for the protograph GLDPC codes, PEG and PMP algorithm. In Section III, we investigate the effect of lifting orders on the local cycle distributions and decoding performance of lifted GLDPC codes. And then, we propose our lifting algorithm, CC-PEG, which makes the concentrated cycles in a form specific to protograph GLDPC codes. Then, we present the numerical results of the CC-PEG codes in Section IV. Finally, Section V proposes some further works and concludes the paper.

II. NOTATIONS AND PRELIMINARIES
In this section, we introduce some notations and preliminaries for protograph-based GLDPC codes, the PMP algorithm for detecting cycles of a given graph, and the PEG algorithm for lifting protograph.

A. PROTOGRAPH GLDPC CODES
In this paper, we consider QC-GLDPC codes lifted from a protograph with parallel edges. The protograph is first lifted by the factor p, which is the maximum number of parallel edges, to remove all parallel edges. Denote the numbers of protograph VNs (PVNs) and CNs in the protograph without parallel edges by n and m. LetV and e denote a set of PVNs and the number of edges in the protograph. The protograph can be represented in terms of an m×n base matrix B = [B i,j ]. By copying the protograph q times and permuting edges, the Tanner graph of the QC-LDPC code is obtained and z = pq is called the lifting factor of the code. Then, the lifted Tanner graph can be expressed in terms of an mq × nq adjacency matrix , where each nonzero entry B i,j is replaced with a q × q circulant permutation matrix (w i,j ) with a shift value w i,j ∈ Z q = {0, 1, . . . , q − 1} and each zero entry is replaced with the q × q zero matrix. Denote the s-th VN, r-th CN in the lifted Tanner graph by v s , c r , and the number of edges by E = eq. Note that the indices for protograph nodes (i, j) and lifted nodes (s, r) are different.
From , component parity check matrices (PCMs) corresponding to GC nodes are required to obtain a full PCM for the GLDPC code. Let c r have an M r × N r component PCM H r . By replacing N r 1's in an r-th row of with columns of H r for all r, an M × N full PCM for the GLDPC code is obtained, where M = mq r=1 M r and N = nq. Each GC node represents the constraint of the (n g , k g ) linear code. Let m g , m s denote the numbers of protograph GC, SPC nodes, respectively. Then, m = m g + m s and the code rate is given as The third term can be viewed as a rate loss caused by doping m g protograph SPC nodes with GC nodes. Denote the sets of doped, undoped PVNs byV d ,V u and denote the sets of doped, undoped VNs in the lifted Tanner graph by V d , V u .

B. THE PMP ALGORITHM
The PMP algorithm [13] is an accurate and efficient method to evaluate the cycle distribution of a given Tanner graph. It can count the exact number of short cycles of length less than 2g, where g denotes the girth of the graph. In the PMP algorithm, the k-th edge e k = (v s , c r ) carries E-tuple vector messages x 2l−1 k , x 2l k at iteration l for k ∈ {1, 2, . . . , E} and l ∈ {1, 2, . . . , g − 1}. First, x 1 k is initialized to the all-zero vector but only one 1 at position k and transmitted from a VN to a CN. Next, for each iteration, all edges pass an extrinsic sum of the previous incoming messages in the opposite direction from the previous iteration.

Meanwhile, each p-th component of x 2l
k indicates the number of length-2l paths from the k-th edge to the p-th edge. Therefore, the k-th component of x 2l+1 k indicates the number of cycles-2l involving the k-th edge. The computational complexity of the PMP algorithm is O(lE 2 ) for iteration l, but it can be easily implemented in parallel with low memory requirements. A simple example of the overall PMP algorithm is depicted in Fig. 2.
To represent the effects of cycles with various lengths on the performance, a metric called the CM was proposed in [13]. The CM of an edge M v s ,c r and the CM of a VN M v s are defined as where N r,s 2l is the number of cycles-2l involving the edge (v s , c r ) and C(v s ) is the set of CNs connected to v s . The parameter β is the constant which is heuristically set to 0.01 in [13], and β l−2 represents the weight of the influence of cycles-2l on the overall graph. In this paper, we also set β = 0.01 and call M v s the VN-CM. If v s has a larger M v s than other VNs, more local cycles containing v s exist. Also, we define the PVN-CM of the j-th PVN as M j = jq s=(j−1)q+1 M v s .

C. THE PEG ALGORITHM AND CYCLE DISTRIBUTION
The PEG algorithm [8] is the most popular method to construct a PCM. In an edge-by-edge manner, the PEG algorithm connects the target VN with a CN in a manner that maximizes the local girth of the target VN. The PEG codes have local optimal girth characteristics leading to excellent performance. The PEG algorithm is also usually adopted to lift a protograph to make a QC-LDPC code [9]. In this case, once the first edge in a certain protograph node is connected, the remaining q − 1 edges are automatically connected based on the QC structure.
However, due to the local greedy behavior of the PEG algorithm, the resulting graphs tend to have many local cycles at the latter lifted side. In Fig. 3, we depict the VN-CMs of regular PEG LDPC codes with VN degree 3. To remove  Fig. 3 shows that the latter lifted VNs have significantly larger VN-CMs. For irregular codes, the PEG algorithm first sorts VNs in ascending order according to the VN degree so that the low degree VNs are located on the former side and have smaller VN-CMs. It places the high degree VNs to the latter lifting order, and thus more local cycles are generated at them. Since cycles containing high degree VNs have a large extrinsic message degree (EMD) and are less harmful to decoding performance [20], the sorting of lifting order by the VN degree gives better decoding performance to PEG codes.

III. THE PROPOSED LIFTING METHOD FOR GLDPC PROTOGRAPH
In the previous section, we verify that the latter processed VNs in the PEG algorithm have more local cycles than the other VNs by comparing VN-CMs. Now, we perform the PEG to lift a protograph GLDPC code, and show that codes with fewer local cycles at vulnerable undoped VNs have significantly better performance than the other codes. Based on this observation, we propose a novel lifting method for protograph GLDPC codes called the CC-PEG algorithm, which makes concentrated cycles at doped VNs and fewer cycles at undoped VNs.

A. EFFECT OF THE CYCLE DISTRIBUTION ON THE PERFORMANCE
In order to show the correlation between the cycle distribution and the performance of GLDPC codes, we compare the three code sets which share the same protograph structure but are lifted with different orders. Each code set consists of 50 distinct codes, generated by using different random seeds for choosing the final CN among the ties. Then, we evaluate the average CMs and performance for each code set.
We consider the regular protograph P 1 of size (n, m) = (21, 9) with VN degree 3 and CN degree 7. One protograph SPC node is replaced by a (7, 4) Hamming GC node. Then, there exist 14 undoped PVNs and 7 doped PVNs. We divide the 14 undoped PVNs into two sets and denote them byV 1 u ,V 2 u , where each set contains 7 PVNs. Then, we lift P 1 using the PEG algorithm with 3 different lifting orders, We denote the three code sets by C F , C M , and C L (subscripts mean the positions ofV d , first, middle, and last). Fig. 4 shows the PVN-CMs for C F , C M , and C L . Since every VN has the same degree, the distribution of PVN-CMs directly reflects the distribution of the local cycles. Similar to Fig. 3, the latter lifted VNs have larger PVN-CMs for all code sets. The last lifted PVN sets,V 2 u ,V 1 u , andV d for C F , C M , and C L , respectively show the largest M j values. If we evaluate the average PVN-CMs of undoped PVNs inV u , the code set C F , C M , and C L show 0.318, 0.370 and 0.270, respectively. In other words, C M has the most local cycles and C L has the least local cycles at vulnerable undoped VNs.
Then, we evaluate the average decoding performance of the code sets C F , C M , and C L over the BEC and AWGN channel. For decoding over the AWGN channel, the iterative belief propagation (BP) decoder uses the standard sum-product algorithm [21] for VNs and SPC nodes. For GC nodes, the decoder calculates bit-wise a posteriori probability and sends it as a message. Similarly, conventional BP and Gaussian elimination operations are performed for decoding over the BEC. The simulated bit error rate (BER) and frame error rate (FER) results are presented in Fig. 5. For both channels, C L shows the lowest error rate, while C M shows the highest error rate. The BER gain resulting from different lifting orders is up to 20 and 3.8 times over the BEC and AWGN channel, respectively. In summary, the code set C L , with the fewest local cycles present at vulnerable undoped VNs, shows the best decoding performance among the three code sets. Through simulation, we confirm that the number of local cycles generated at undoped VNs has a significant impact on the decoding performance of GLDPC codes.

B. THE CC-PEG ALGORITHM FOR PROTOGRAPH GLDPC CODES
Inspired by the previous observations, we propose a new lifting method called the CC-PEG algorithm for protograph GLDPC codes. First, we consider the lifting order of the VNs. For regular protographs, it is shown that placing doped VNs 57288 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. to the latter order makes fewer local cycles at undoped VNs, resulting in better decoding performance. However, for the case of irregular protographs, the lifting order of VNs should be sorted in ascending order by the VN degree to maximize the EMD characteristics of the cycles. Through extensive simulations, we find a superior ordering policy: i) First, sort out VNs in the degree ascending order, and then ii) place the doped VNs backward among the VNs with the same degree. As well-constructed protographs typically have higher-degree doped VNs than undoped VNs [15], [16], most doped VNs are still arranged at the back side in accordance with this ordering policy.
Next, we introduce the edge connection strategy that aims to concentrate local cycles at doped VNs and generates fewer local cycles at undoped VNs. The first selection criterion of the CC-PEG is the length of the shortest cycle which contains at least one undoped VN. We call this the undoped girth and denote it by g u . To calculate g u , we utilize the PMP algorithm. We can easily find the length of the shortest cycle including a certain edge by the PMP process without additional computation. In detail, g u is twice of the minimum l which satisfies N r,s 2l ̸ = 0, v s ∈ V u . Note that,

Algorithm 1
The CC-PEG Algorithm Input: B m×n , q, V d , V u , α, β Output: mq×nq 1: Initialize ← 0 (i.e. all w i,j ← −1) 2: for j = 0 to n − 1 do 3: for i = 0 to m − 1 do 4: if B i,j ̸ = 0 then 5: for t = 0 to q − 1 do 6: w i,j ← t 7: perform the PMP algorithm [13] with 8: end if 15: end for 16: end for although the most harmful cycles are the cycles that only contain undoped VNs (denote the shortest length of these cycles by g U ), we find that the performances of the lifted code sets considering g u and g U are similar. Therefore, we choose g u instead of g U as our first criterion because the evaluation of g u is much easier than that of g U .
The second criterion of the CC-PEG is the concentrated CM M cc defined as where α is a positive constant less than 1. Among the edges having the largest g u equally, the CC-PEG chooses the final candidates resulting in the lowest M cc . According to the definition of M cc , local cycles at undoped VNs in V u are treated α −1 times more harmful than those at doped VNs in V d . A small α value makes fewer local cycles at undoped VNs, but it may lead to excessive cycles at doped VNs. If there are multiple candidates even after the second selection, the CC-PEG randomly picks one edge among the candidates. The proposed CC-PEG algorithm is described in Alg. 1. Note that the argmax and argmin operations in lines 11 and 12 return a set of arguments if there are multiple candidates with the same maximum or minimum value.
Since the computational complexity of the PMP algorithm is O(lE 2 ) for iteration l, the total complexity of the CC-PEG algorithm is O(ql 1 q 2 ) + O(ql 2 (2q) 2 ) + · · · + O(ql e (eq) 2 ) ≈ O(l max (e(e + 1)(2e + 1)/6)q 3 where l i is the number of PMP iterations in the process of selecting the i-th shift value, and l max is the maximum of l i 's. We set the limit of l max to 10 to prevent excessive search (ignore cycles longer than 20). The complexity of the VOLUME 11, 2023 57289 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. CC-PEG is about one-third of the QC-PMP [13] algorithm (although their complexity is the same in the big O notation), which initializes all w i,j randomly and construct the fully lifted from the start of the algorithm. Meanwhile, since the computational complexity of one PEG edge selection process is O(M ) [8], the complexity of the PEG lifting algorithm [9] is approximately O(eM ) ≈ O(e 2 q). Therefore, the CC-PEG algorithm can be considered significantly more complex than the PEG algorithm. However, this is not detrimental to the channel coding system as the proposed CC-PEG algorithm is a code constructing technique. Once the code has been successfully constructed, the complexity of the CC-PEG algorithm has no further impact on the communication system. In the following simulations, we confirm that the CC-PEG algorithm is computationally feasible for constructing short to moderate length codes, where GLDPC codes have shown a performance advantage over LDPC codes.

IV. NUMERICAL RESULTS AND ANALYSIS
In this section, we investigate the cycle distribution and compare the decoding performances of GLDPC codes lifted by the PEG [9], QC-PMP [13], and CC-PEG. Since there has been no research conducted on lifting methods specially tailored for GLDPC codes, we choose the most commonly used lifting algorithms, PEG and QC-PMP, for comparison. The regular protograph P 1 and irregular protographs (16-1),  in [15], and (6), (5) in [16], which show near-capacity asymptotic performances, are employed to demonstrate the effectiveness of the CC-PEG algorithm over various protographs, lengths, and code rates. Denote the irregular protographs by P 2 -P 5 in the order written above. Note that they are arranged according to the numbers of edges in the protographs, which are 21, 24, 47 and 69. Again, we evaluate the average values of girth metrics, CMs, and decoding performances for 50 codes.
To confirm the cycle concentration of CC-PEG codes, Fig. 6 shows the PVN-CMs of GLDPC codes lifted by different methods. The conventional PEG algorithm displays an increasing PVN-CM trend as the lifting process progresses, as shown in Fig. 4. Compared with the PEG algorithm, the CC-PEG algorithm with α = 1 generates GLDPC codes with fluctuating local cycles. When α = 0.01 is used, extremely low PVN-CMs are generated at undoped VNs, but excessive cycles are generated at doped VNs. On the other hand, α = 0.1 produces affordable local cycles for doped VNs, while still maintaining low PVN-CMs for most undoped VNs. The QC-PMP algorithm optimizes cycles uniformly over all VN regions [13], and exhibits evenly distributed small cycles across all PVNs, as intended. However, QC-PMP does not prioritize the cycle length in the undoped VN region and generates codes with worse decoding performance than other algorithms, as we will elaborate in the following section. Table 1 illustrates the total VN-CMs at undoped VNs M V U and doped VNs M V D for various α values, lifted from P 3 . As α decreases, the tendency to concentrate cycles toward doped VNs intensifies, resulting in lower M V U and higher M V D values, as intended. However, for α ≤ 0.02, M V D increases excessively, leading to a degradation in the actual decoding performance. Conversely, when α ≥ 0.5, M V U does not decrease sufficiently, which was also found to negatively affect decoding performance. Across various protographs, α = 0.05 ∼ 0.2 produces codes that best meet our intended purpose and exhibit superior performance compared to existing algorithms. Therefore, we fix α at 0.1 for our following experiments. Fig. 7 illustrates the BER and FER curves of GLDPC codes lifted from the regular protograph P 1 by the QC-PMP [13], PEG [9], and CC-PEG algorithm. Over the BEC, all three code sets exhibit similar performance in the range of FER ≥ 10 −4 , but CC-PEG codes outperform the other codes in the range of FER < 10 −4 , showing a lower error floor. Over the AWGN channel, the CC-PEG codes also show the best performance with a FER gain of more than 0.2 dB compared to the QC-PMP codes at FER = 2 × 10 −5 . Note that the scales of the y-axis are different between the two channels due to the long simulation time required for bit-wise APP decoding in GC nodes over the AWGN channel.

A. DECODING PERFORMANCE EVALUATION
Next, we apply the CC-PEG algorithm to the irregular protograph with capacity-approaching performance, which has more practical meaning than regular one. In Fig. 8, the decoding performances of the GLDPC codes lifted from P 2 and P 3 [15] are depicted. In these cases, the numbers of 57290 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.  edges in protographs are relatively small and search spaces of the lifting algorithms are limited, which do not allow for significant differences among the algorithms. However, even here, the CC-PEG codes show similar or better performances compared to the other codes. For P 3 , which has slightly more edges than P 2 , the CC-PEG codes demonstrate a clear perfor-mance advantage, particularly with z = 60 over the AWGN channel. At FER = 3 × 10 −4 , it achieves a gain of about 0.1 dB compared to the PEG codes, and 0.2 dB compared to the QC-PMP codes.
The CC-PEG codes lifted from the larger protographs P 4 and P 5 [16] with more edges show much larger gains, as shown in Fig. 9. For P 4 , the CC-PEG codes achieve ϵ = 0.03 gain compared to the PEG codes at FER = 10 −3 for both z = 30 and z = 60 over the BEC. Also, the CC-PEG codes gain 0.25 dB and 0.4 dB for z = 30 and z = 60, respectively at FER = 4 × 10 −4 over the AWGN channel. The codes lifted from P 5 show the largest performance gap over the BEC. Here, the CC-PEG codes with z = 30 gain more than ϵ = 0.05 at FER = 10 −4 , which show the 20 times FER difference at ϵ = 0.20. And the CC-PEG codes with z = 60 gain about 0.25 dB at FER = 10 −3 over the AWGN channel. Further, it is expected that the gains of the CC-PEG codes would be much larger in the high SNR region.
In Figs. 7-9, note again that the all codes in the same figure are constructed from the same protograph, which means that they have the same asymptotic performance. Thus, the performance gaps purely come from the different lifting methods. The constructed codes have short to moderate code lengths, wherein GLDPC codes exhibit superior performance compared to LDPC codes [5], hence possessing more practicality. The CC-PEG codes show 1-20 times FER gains in all cases, and the gains become greater when the given protograph has more edges, providing a larger search space for the lifting algorithm to choose from.
Note that QC-PMP codes were known to outperform PEG codes for LDPC codes [13], but the lack of local greedy behavior makes the performances of the QC-PMP codes worse than the PEG codes for GLDPC codes. This result also supports the strategy of the CC-PEG algorithm that deliberately makes more local greedy behavior to get a better performance for GLDPC codes. We provide some evidences to prove this claim in the following section.

B. GIRTH AND CYCLE METRICS OF CC-PEG CODES
Since the CC-PEG codes and comparison codes share the same protograph structure, theoretical analysis methods such as density evolution [19] cannot explain the gain achieved by CC-PEG codes. Therefore, we evaluate several quantitative metrics related to the decoding performance of GLDPC codes to analyze the reasons for the performance improvement achieved by CC-PEG codes. Table 2 presents the average girth and cycle metrics of GLDPC code sets lifted from P 2 -P 5 . The lifting factors of the code sets are z = 120, 120, 60, 60. In the table, g u denotes the undoped girth, and the M V U (M V D ) is the total VN-CMs at undoped (doped) VNs. To evaluate the most harmful cycles, which consist only of undoped VNs, we introduce two additional metrics, g U and M U T . They denote the girth and the total CM of the subgraph consisting only of undoped VNs. The CC-PEG codes show the largest g u , g U , and the smallest M V U , M U T for most protographs. This indicates that the CC-PEG successfully achieves a large girth and a small number of local cycles at vulnerable undoped variable nodes by concentrating cycles on doped variable nodes. On the other hand, the PEG codes present large g U because the PEG has the local optimal characteristic, thus making large local girth at the former lifted undoped VNs. However, they show weakness in the number of cycles at undoped VNs. Meanwhile, the QC-PMP codes have a small number of cycles over all VNs (smaller M V U , M V D ), but show small g U because it treats the cycles of all areas equally and does not give an advantage to the former lifted VNs.

V. CONCLUSION
By placing the undoped VNs to the former lifting order, the number of the local cycles at vulnerable undoped areas could be reduced and the resulting codes showed better performances. To further enhance performance through adjusting the cycle distribution, we proposed optimal VN lifting orders of protograph GLDPC codes and the new lifting algorithm called the CC-PEG, designed specially for GLDPC codes. The CC-PEG algorithm, which locally optimizes the undoped girth and proposed CM, concentrated local cycles to doped VNs and make fewer local cycles at undoped VNs. Extensive simulations confirmed the superiority of CC-PEG codes for various protographs, exhibiting up to 20 and 5 times FER gain over the BEC and AWGN channel, respectively. To support the superiority of CC-PEG codes, we proposed several girth and cycle metrics related to GLDPC structure. The metrics demonstrate that the constructed CC-PEG codes have large local girth and the small number of local cycles at vulnerable undoped VNs.