# On the Construction of Protograph-based Partially Doped GLDPC Codes

Jaewha Kim, Jae-Won Kim, and Jong-Seon No J. Kim is with the Department of
Electrical and Computer Engineering, INMC, Seoul National University, Seoul
08826, Korea (e-mail: woghk9307@snu.ac.kr).J.-W. Kim is with the Department of
Electronic Engineering, Engineering Research Institute (ERI), Gyeongsang
National University, South Korea (e-mail: jaewon07.kim@gnu.ac.kr).J.-S. No
with the Department of Electrical and Computer Engineering, INMC, Seoul
National University, Seoul 08826, Korea (e-mail: jsno@snu.ac.kr).J.-W. Kim is
the corresponding author.

###### Abstract

A generalized low-density parity-check (GLDPC) code is a class of codes, where
single parity check nodes in a conventional low-density parity-check (LDPC)
code are replaced by linear codes with higher parity check constraints. In
this paper, we introduce a new method of constructing GLDPC codes by inserting
the generalized check nodes for partial doping. While the conventional
protograph GLDPC code dopes the protograph check nodes by replacing them with
the generalized check nodes, a new GLDPC code is constructed by adding the
generalized check nodes and partially doping the selected variable nodes to
possess higher degrees of freedom, called a partially doped GLDPC (PD-GLDPC)
code. The proposed PD-GLDPC codes can make it possible to do more accurate
extrinsic information transfer (EXIT) analysis and the doping granularity can
become finer in terms of the protograph than the conventional GLDPC code. We
also propose the constraint for the typical minimum distance of PD-GLDPC codes
and prove that the PD-GLDPC codes satisfying this condition have the linear
minimum distance growth property. Furthermore, we obtain the threshold
optimized protograph for both regular and irregular ensembles of the proposed
PD-GLDPC codes over the binary erasure channel (BEC). Specifically, we propose
the construction algorithms for both regular and irregular protograph-based
PD-GLDPC codes that enable the construction of GLDPC codes with higher rates
than the conventional ones. The block error rate performance of the proposed
PD-GLDPC code shows that it has a reasonably good waterfall performance with
low error floor and outperforms other LDPC codes for the same code rate, code
length, and degree distribution.

###### Index Terms:

Binary erasure channel (BEC), doping, generalized low-density parity-check
(GLDPC) code, low-density parity-check (LDPC) code, partially doped GLDPC (PD-
GLDPC) code, protograph, protograph extrinsic information transfer (PEXIT),
typical minimum distance.

##  I Introduction

Low-density parity-check (LDPC) codes, first introduced in [1], have received
many interests due to their low decoding complexity by the sparsity of the
parity check matrix and Shannon capacity approaching error performance [2]. An
LDPC code is defined over a bipartite graph consisting of variable nodes and
single parity check (SPC) constraint nodes. A generalized LDPC (GLDPC) code
was first introduced in [3], which is constructed by replacing (doping) some
SPC nodes by generalized constraint (GC) nodes in a parity check matrix (PCM)
of a linear code with a larger minimum distance [4]. This makes the LDPC code
to have a larger minimum distance [5], faster decoding convergence [6], and a
better asymptotic threshold at the cost of using additional decoding
complexity and redundancy [7].
Many types of linear codes for GC nodes, also called as the component codes,
are used in the GLDPC codes such as Hamming codes [8], Hadamard codes [9],
Bose–Chaudhuri–Hocquenghem (BCH) codes, and Reed-Solomon (RS) codes [10]. The
research of the GLDPC codes is extended to spatially coupled LDPC (SC-LDPC)
codes [11, 12, 13] and doubly GLDPC (DGLDPC) codes [14, 15, 16]. While some
capacity-approaching GLDPC codes were constructed using irregular random GLDPC
codes [7, 17], there are limited works related to the capacity-approaching
protograph-based GLDPC codes due to two difficulties of the code construction.
First, since the rate-loss that is incurred during the doping process is not
negligible, constructing a protograph GLDPC code for a medium to high code
rate is difficult. Second, due to the high doping granularity, the
optimization of protograph GLDPC codes is limited. In order to replace a
specific SPC node with a GC node, the degree of the SPC node and length of the
component code should be the same. Thus, the selection of SPC nodes to be
replaced becomes limited, which makes the doping process difficult.
In this paper, we propose a new construction method of protograph-based GLDPC
code, called partially doped GLDPC (PD-GLDPC) codes, which resolves the
limitations for the construction of the conventional protograph GLDPC codes.
In the proposed construction, constraint rows for a GC node are added, rather
than replaced by partially doping the selected variable nodes, to the parity
check matrix after the lifting process of a protograph. Furthermore, we
propose an optimized construction method of the protograph-based PD-GLDPC
codes while the newly constructed code has a minimum distance that grows
linearly with the block length of the code from the constraint for the
existence of the typical minimum distance. Both the new optimized code
construction method and its analysis of the proposed protograph-based PD-GLDPC
codes are proposed in this paper, which makes the proposed protograph-based
PD-GLDPC code approach the channel capacity.
Specifically, by focusing the partial doping on the protograph variable nodes
of degree 2, we construct protograph-based PD-GLDPC codes with a good
waterfall performance. Furthermore, the proposed protograph-based PD-GLDPC
codes have more accurate protograph extrinsic information transfer (PEXIT)
analysis compared to conventional protograph GLDPC codes, which is due to the
exact computation of the extrinsic information transfer (EXIT) for a
protograph GC node in the proposed protograph-based PD-GLDPC codes. While a GC
node in conventional protograph GLDPC codes computes the average mutual
informations of its neighboring protograph variable nodes, a GC node in the
proposed protograph-based PD-GLDPC codes computes the mutual information of
single protograph variable node as its neighborhood. Thus, the computation of
the PEXIT of a GC node becomes more accurate in the proposed PD-GLDPC codes.
Since the proposed protograph-based PD-GLDPC codes are constructed through the
doping process by adding new GC nodes of the error correcting codes connected
to the subset of variable nodes lifted from single protograph variable node,
the code construction can be made over any base matrix, regardless of the
check node degree distribution of the original protograph. That is, the new
rows for the doped GC nodes are added by the component code after lifting,
where the lifting size is the multiple of the length of the component code and
the variable nodes in each added row are lifted from single protograph
variable node. Thus, the doping granularity can be much smaller than the
conventional GLDPC codes, which enables much flexible construction of the
proposed PD-GLDPC codes.
Finally, we propose the optimization method to construct large protographs for
the proposed protograph-based PD-GLDPC codes. Due to the large rate loss from
protograph GC nodes, it is difficult to construct GLDPC codes with medium to
high code rates from small protographs. The large protographs of high code
rates constructed from the degree distribution based on the random ensemble
enable the larger number of doping of GC nodes. Also, the number of doping of
GC nodes can be varied according to the code rate of the initial protograph.
The optimized balance between the number of doping nodes and the code rate of
the initial protograph is given in the proposed protograph-based PD-GLDPC
code. Furthermore, the construction of the proposed protograph-based PD-GLDPC
codes with code rates $1/2$ and $1/4$ are presented in this paper.
The rest of the paper is organized as follows. In Section II, we introduce
some preliminaries on the binary erasure channel (BEC) and protograph GLDPC
codes. Section III illustrates the proposed protograph-based PD-GLDPC code
structure and derives its PEXIT analysis with performance comparison with the
conventional GLDPC codes. The construction and optimization of protograph-
based PD-GLDPC codes from regular and irregular protographs are given in
Section IV. Section V shows the optimized results and error performance of the
proposed codes. Section VI concludes the paper with some discussion of the
results.

##  II Backgrounds

In this section, we introduce some notations and concepts of a binary erasure
channel, protograph LDPC codes, and the construction method of conventional
protograph GLDPC codes. The EXIT analysis and the decoding process of
conventional GLDPC codes are also briefly introduced.

###  II-A Protograph LDPC Code and BEC

Let $\mathbb{x}=\\{x_{1},{\cdots},x_{k}\\},x_{i}\in\\{0,1\\}$ be a $k$-bit
binary message vector, which is encoded via an $(n,k)$ linear code, forming an
$n$-bit codeword $\mathbb{c}=\\{c_{1},{\cdots},c_{n}\\},c_{i}\in\\{0,1\\}$.
The codeword passes through a memoryless BEC, where each bit is either erased
with probability $\epsilon$ or correctly received.
A protograph LDPC code is defined by a relatively small bipartite graph
$G=(V,C,E)$ representing a protograph, where
$V=\\{v_{1},{\cdots},v_{n_{v}}\\}$ is a set of variable nodes and
$C=\\{c_{1},{\cdots},c_{n_{c}}\\}$ is a set of check nodes. Let $E$ be the set
of edges $e$, where $e=(v,c)$ connects the variable node $v\in V$ and the
check node $c\in C$. The bipartite graph can also be expressed in terms of an
$n_{c}\times n_{v}$-sized base matrix ${\mathbb{B}}_{n_{c}\times
n_{v}}=\\{b_{i,j}\\},i\in[n_{c}],j\in[n_{v}]$, where
$b_{i,j}\in\\{0,1,2,{\cdots}\\}$ and $[A]$ is a set of positive integers less
than or equal to a positive integer $A$. The rows represent the check nodes
and the columns represent the variable nodes in the protograph. Each entry
$b_{i,j}$ of the base matrix represents the number of edges connected between
a variable node and a check node. Parallel edges are allowed and if there are
no edges connected between $v_{j}$ and $c_{i}$, the entry $b_{i,j}$ is zero.
The variable (check) node degree $deg(v_{j})$ ($deg(c_{i})$) is defined as the
number of edges incident to itself. A protograph LDPC code [19] is constructed
by copy-and-permute operation of $G$. The bipartite graph $G$ is copied by the
lifting factor $N$ and the copies of each edge $e=(v,c)\in E$ are permuted
among the copies of $v$ and $c$. In general, the large value of $N$ guarantees
the sparseness of the code.

###  II-B Construction of Protograph GLDPC Code [18]

Conventionally, a protograph GLDPC code is constructed by replacing (doping) a
check node of a protograph with a GC node that has a parity check constraint
from an $(n_{i},k_{i},d_{min}^{i})$ linear code (component code), where
$n_{i}$ ($k_{i}$) is the code length (dimension) and $d_{min}^{i}$ is the
minimum distance of the component code for a check node $c_{i}$. The condition
for replacement is that the check node degree should be exactly equal to the
length of the component code, i.e., $deg(c_{i})=n_{i}$. Note that the original
check node has the parity check constraint of an
$(n_{i},k_{i})=(deg(c_{i}),deg(c_{i})-1)$ SPC code. The code rate $R$ of
protograph GLDPC code is $R=1-\frac{m_{proto}}{n_{v}}$, where
$m_{proto}=\sum_{i=1}^{n_{c}}(n_{i}-k_{i})$. While the minimum distance of an
SPC node is 2, the variable nodes connected to the GC node will be protected
by parity check constraints of the component code with the minimum distance
larger than two. Fig. 1 is the conventional protograph GLDPC code of code rate
$\frac{3}{7}$ by replacing an SPC node with the $(7,4)$ Hamming code.

Figure 1: Conventional protograph GLDPC code construction by replacing an SPC
node with a protograph GC node using the $(7,4)$ Hamming code as the component
code.

###  II-C PEXIT Analysis and Decoding Process of Protograph GLDPC Code

Algorithm 1 The PEXIT analysis of a protograph GLDPC code [20]

Step 1) Initialization
Initialize $I_{ch}(j)=\epsilon$ for $j\in[n_{v}]$.
Step 2) Message update from variable node to check node
Update $I_{EV}(i,j)=1-\epsilon\prod_{t\in N(v_{j}),t\neq
i}\bigl{(}1-I_{AV}(t,j)\bigr{)}$ for all $j\in[{n_{v}}]$, where
$I_{EV}(i,j)=0$ if $b_{i,j}=0$. If $c_{i}$ is an SPC node,
$I_{AV}(i,j)=I_{EC}(i,j)$ and if $c_{i}$ is a GC node,
$I_{AV}(i,j)=I_{EGC}(i,j)$.
Step 3) Message update from check node to variable node
For all $i$, if $c_{i}$ is an SPC node, go to Step 3-1) and if $c_{i}$ is a GC
node, go to Step 3-2).
Step 3-1) $I_{EC}(i,j)=\prod_{t\in N(c_{i}),t\neq j}I_{AC}(i,t)$, where
$I_{AC}(i,t)=I_{EV}(i,t)$.

Step 3-2) For all $j\in$ $N(c_{i})$, compute

$I_{EGC}(i,j)=\frac{1}{n_{i}}\sum_{h=1}^{n_{i}}\bigl{(}1-I_{AGC}(i)\bigr{)}^{h-1}\bigl{(}I_{AGC}(i)\bigr{)}^{n_{i}-h}[h\tilde{e}_{h}-(n_{i}-h+1)\tilde{e}_{h-1}],$
(1)

where $I_{AGC}(i)=\frac{1}{n_{i}}\sum_{j=1}^{n_{i}}I_{EV}(i,j)$.
Step 4) APP mutual information computation
For all $j\in[n_{v}],I_{APP}(j)=1-\epsilon\prod_{t\in
N(v_{j})}\bigl{(}1-I_{AV}(t,j)\bigr{)}$. If $c_{t}$ is an SPC node,
$I_{AV}(t,j)=I_{EC}(t,j)$ and if $c_{t}$ is a GC node,
$I_{AV}(t,j)=I_{EGC}(t,j)$.
Step 5) Convergence check of variable nodes
Repeat Step 2)–4) until $I_{APP}(j)=1,$ for all $j\in[n_{v}]$.

The asymptotic performance of the conventional protograph GLDPC codes is
evaluated by the PEXIT analysis. We use the EXIT and PEXIT analyses of the
GLDPC code over the BEC introduced in [20, 21, 22]. The PEXIT analysis tracks
down the mutual information of extrinsic messages and a priori error
probabilities of the variable nodes, SPC check nodes, and GC nodes of a
protograph GLDPC code. Let $I_{ch}(j)$ be the channel information from the
erasure channel for the protograph variable node $v_{j}$. In addition,
$I_{EV}(i,j)$ $(I_{EC}(i,j))$ is the extrinsic information sent from $v_{j}$
($c_{i}$) to $c_{i}$ ($v_{j}$) and $I_{AV}(i,j)$ $(I_{AC}(i,j))$ is the a
priori mutual information of $v_{j}$ ($c_{i}$) sent from $c_{i}$ ($v_{j}$),
where $c_{i}$ is an SPC node. For GC nodes, we use the notations $I_{AGC}(i)$
and $I_{EGC}(i,j)$ for a priori and extrinsic information. Let $N(c_{i})$
($N(v_{j})$) be the set of variable (check) nodes incident to $c_{i}$
($v_{j}$), i.e., neighborhood of $c_{i}$ ($v_{j}$). Finally, $I_{APP}(j)$ is a
posteriori probability of $v_{j}$. To explain (1), if $c_{i}$ is a GC node
with the $(n_{i},k_{i})$ Hamming code, the PEXIT of the GC node is computed
from a closed form using the property of the simplex code, which is the dual
code of a Hamming code. Also,
$I_{AGC}(i)=\frac{1}{n_{i}}\sum_{j=1}^{n_{i}}I_{EV}(i,j)$ is the average a
priori mutual information for a GC node to compute the PEXIT message. In (1),
we have

$\tilde{e}_{h}=\sum_{t=1}^{h}t\sum_{u=0}^{t-1}(-1)^{u}2^{\binom{u}{t}}{k_{i}\brack
t}{t\brack u}\binom{2^{t-u}}{h}.$

For two positive integers $a$ and $b$, we also have
$\binom{a}{b}=\prod_{i=0}^{b-1}\frac{a-i}{b-i}$ and ${a\brack
b}=\prod_{i=0}^{b-1}\frac{2^{a}-2^{i}}{2^{b}-2^{i}}$, where $\binom{a}{0}=1$
and ${a\brack 0}=1$. The PEXIT process is given in Alg. 1, which searches for
the minimum $\epsilon$ to successfully decode, i.e., $I_{APP}(j)=1,$ for all
$j\in[n_{v}]$, in asymptotic sense.
Now, we briefly explain the decoding process of the conventional GLDPC codes
[13]. The variable nodes process the conventional message-passing decoding
over the BEC by sending correct extrinsic messages to the check nodes if any
of the incoming bits from its neighborhood is not erased. The SPC nodes send
correct extrinsic messages to the variable nodes if all of its incoming
messages are correctly received, and send an erasure message otherwise. In
this paper, the decoding of GC nodes is processed by the maximum likelihood
(ML) decoder. For each iteration, a GC node $c_{i}$ with the $(n_{i},k_{i})$
component code receives the set of erasure locations $\\{e_{i}\\}$ from
$N(c_{i})$. Let $H_{GC}$ be the parity check matrix of the component code and
$H_{e}$ be the submatrix of $H_{GC}$ indexed with $\\{e_{i}\\}$. The decoder
computes the Gaussian-elimination operation of $H_{e}$, making it into a
reduced row echelon form $H_{e}^{reduced}$. If
rank($H_{e}^{reduced}$)$=|\\{e_{i}\\}|$, the GC node solves all the input
erasures and otherwise, the decoder corrects the erasures corresponding to the
rows with weight 1 from $H_{e}^{reduced}$. The decoding complexity can be
further reduced if the GC node exploits bounded distance decoding; however,
the degradation of asymptotic performance is not negligible, as shown in [7].

##  III The Proposed Protograph-based PD-GLDPC Code

(a)

(b)

(c)

Figure 2: Block diagram of the construction process of conventional GLDPC
codes and the proposed protograph-based PD-GLDPC codes.

In this section, a new construction method of protograph-based GLDPC codes is
proposed. While the conventional protograph GLDPC codes are constructed by
replacing some protograph SPC nodes in the original protograph by GC nodes
using the component code, the proposed protograph-based PD-GLDPC code is
constructed by adding the GC nodes for the subset of variable nodes using
component codes after the lifting process of the original protograph, where
each GC node is connected to the variable nodes copied from single protograph
variable node. A block diagram of the construction process of both codes
together with the conventional random GLDPC code is given in Fig. 2.

###  III-A Construction Method of Protograph-based PD-GLDPC Code

Figure 3: An example of a proposed $({\mathbb{B}}_{2\times 3},7,4,14,1)$
protograph-based PD-GLDPC code construction, where ${\mathbb{B}}_{2\times
3}=[1\,1\,1;1\,0\,1]$ and $\pi$ is the $14\times 14$ sized permutation matrix.

First, we define the partial doping for variable nodes using the addition of
GC nodes to a lifted protograph, that is, the addition of rows for the parity
check matrix by the component code, where each GC node is incident to variable
nodes copied from single protograph variable node. Also, we define a partially
doped protograph variable node as a protograph variable node incident to the
added GC nodes. While the term doping in conventional GLDPC codes is used in
the perspective of check nodes, we use the term partial doping in the
perspective of variable nodes. Let ${\mathbb{B}}_{n_{c}\times n_{v}}$ be an
$n_{c}\times n_{v}$ base matrix, where some protograph variable nodes are
partially doped with a $(\mu,\kappa)$ component code after the lifting
process. Let $x=0,1,2,{\cdots}$ be the number of the partially doped
protograph variable nodes, where each protograph variable node is doped by
$N/\mu$ component codes after the lifting process. Then, a proposed
protograph-based PD-GLDPC code is defined with parameters
$({\mathbb{B}}_{n_{c}\times n_{v}},\mu,\kappa,N,x)$. We assume that the
component code used in the paper is the $(\mu,\kappa)$ Hamming code and that
$\mu$ divides the lifting factor $N$ such that $N=\mu\beta$, where $\beta$ is
a non-negative integer. The variable nodes copied from $x$ protograph variable
nodes in the base matrix are partially doped by the GC nodes. That is, in the
proposed PD-GLDPC code construction, the $N/\mu$ GC nodes are connected to the
$N$ variable nodes lifted from each protograph variable node. Thus, the
proposed construction method can choose the protograph variable nodes to
protect by partial doping. An example of the proposed construction is given in
Fig. 3, which illustrates the doping process by a $(7,4)$ component Hamming
code over a $2\times 3$ base matrix.

(a)

(b)

Figure 4: PCM of a protograph-based PD-GLDPC code.

The basic concept of the proposed construction is to focus on the protection
of the variable nodes lifted from a protograph variable node. Since $N$ is the
multiple of the component code length, all the variable nodes lifted from
single protograph variable node can be protected by using $\beta$ GC nodes.
The parity check matrix for $\beta$ GC nodes, connected to the variable nodes
lifted from a protograph variable node, $\textbf{H}_{\textbf{H}}$ is shown in
Fig. 4(a), where $\textbf{H}_{Hamm}$ is the parity check matrix of the Hamming
code. The constructed protograph-based PD-GLDPC code has a PCM
${\mathbb{H}}_{PD{\text{-}}GLDPC}$ as in Fig. 4(b), where the upper part is
the PCM of the added $\beta x$ GC nodes and the lower part
${\mathbb{H}_{proto}}$ refers to the PCM of the LDPC code lifted from the
original protograph. Since the doping proceeds after the lifting process, a
protograph-based PD-GLDPC code cannot be expressed in terms of a protograph.
Although for the proposed construction method, the number of partial doping
doesn’t necessarily have to be a multiple of $\mu$, in order to make the
number of added rows the multiple of the lifting size, similar to the size of
the conventional one, we partially dope the protograph variable nodes by a
granularity $\mu$, i.e., $x=0,\mu,2\mu,3\mu,{\cdots}$. Assuming that $x$ is a
multiple of $\mu$, let $y$ be a non-negative integer representing the number
of bulks of doped protograph variable nodes, where each bulk is a set of
protograph variable nodes with size $\mu$, i.e., $y=\frac{x}{\mu}$.

###  III-B The PEXIT Analysis of Protograph-based PD-GLDPC Codes

Figure 5: An example of the Tanner graph representation of the proposed PD-
GLDPC code from the protograph ${\mathbb{B}}_{2\times 3}=[1\,1\,1;1\,0\,1]$,
where $x=1$.

The PEXIT of the proposed protograph-based PD-GLDPC codes is similar to that
of the conventional protograph GLDPC codes in Alg. 1 except for the EXIT of a
GC node. Since the incoming mutual information of each GC node is obtained
from only one protograph variable node in the proposed code, the average
mutual information sent to each GC node is the same as the extrinsic message
of the protograph variable node connected to the GC node. Let $b_{j},j\in[x]$
be the virtual node representing the set of $\beta$ GC nodes connected to the
protograph variable node $v_{j}$. An example of the representation of a
virtual node over a protograph is given in Fig. 5. Note that although $b_{j}$
is not a protograph node itself, it is possible to compute the PEXIT of a
protograph-based PD-GLDPC code. Also, let $I_{EV}^{(b_{j})}(j)$ be the
extrinsic information from $v_{j}$ to $b_{j}$ expressed as

$I_{EV}^{(b_{j})}(j)=1-\epsilon\prod_{t\in N(v_{j})}(1-I_{AV}(t,j)),j\in[x].$

Since $b_{j}$ is solely connected to $v_{j}$, the index term for the extrinsic
information from $v_{j}$ to $b_{j}$ is expressed by the notation of $j$ only.
In order to compute the EXIT of $b_{j}$, let $I_{AGC}^{(b_{j})}(j)$ and
$I_{EGC}^{(b_{j})}(j)$ be the a priori and extrinsic mutual informations of
$b_{j}$, respectively. Note that the EXIT of each GC node is computed from
single a priori mutual information to process single value of extrinsic mutual
information for the neighboring variable nodes. For the conventional
protograph GLDPC codes, the a priori mutual information of a protograph GC
node is computed by averaging the extrinsic mutual information of its
neighboring variable nodes. However, for the proposed protograph-based PD-
GLDPC codes, since $b_{j}$ receives the extrinsic mutual information of
$v_{j}$ only, it is clear that
$I_{AGC}^{(b_{j})}(j)=I_{EV}^{(b_{j})}(j),\,j\in[x]$. We also compute the
extrinsic mutual information from $b_{j}$ to $v_{j}$ denoted as
$I_{EGC}^{(b_{j})}(j)$ using (1), given the a priori mutual information
$I_{AGC}^{(b_{j})}(j)$, which is given as

$I_{EGC}^{(b_{j})}(j)=\frac{1}{\mu}\sum_{h=1}^{\mu}\bigl{(}1-I_{AGC}^{(b_{j})}(j)\bigr{)}^{h-1}\bigl{(}I_{AGC}^{(b_{j})}(j)\bigr{)}^{\mu-h}[h\tilde{e}_{h}-(\mu-h+1)\tilde{e}_{h-1}].$
(2)

Note that for the proposed protograph-based PD-GLDPC codes, the a priori
(extrinsic) EXIT of the GC node is computed from the extrinsic (a priori) EXIT
of single protograph variable node. While the EXIT of variable nodes and SPC
nodes for the proposed protograph-based PD-GLDPC codes is the same as that of
the conventional protograph GLDPC codes described in Alg. 1, the EXIT of the
GC nodes in the proposed codes is changed to (2) whereas the conventional
protograph GLDPC codes use (1).
In general, as the portion of degree-2 variable nodes in the LDPC codes
increases, the asymptotic performance is enhanced [23], but their minimum
distance decreases and then the error floor becomes worse. Although the
proposed protograph-based PD-GLDPC code construction method enables the
partial doping for any variable nodes in the given protographs, we focus only
on partial doping for degree-2 variable nodes as follows. First, we construct
the original base matrix ${\mathbb{H}_{proto}}$ with the large portion of
degree-2 variable nodes. Then, we partially dope some of the protograph
variable nodes of degree-2 to increase the minimum distance and improve their
performance. Thus, regular protographs of degree-2 variable nodes and
irregular protographs with many degree-2 variable nodes are used for the
construction of the proposed protograph-based PD-GLDPC codes. In terms of
irregular ensemble LDPC codes, a large portion of degree-2 variable nodes
enables the LDPC code to achieve the capacity approaching performance [24]. On
the other hand, by reasonably selecting the number of partially doped variable
nodes for degree-2, the property of the linear minimum distance growth with
the length of the LDPC code can be guaranteed. Thus, when we design the
proposed protograph-based PD-GLDPC codes, balancing the partial doping over
degree-2 variable nodes enables both the existence of a typical minimum
distance and a good asymptotic performance.

###  III-C Condition for Typical Minimum Distance of Protograph-based PD-GLDPC
Code

The existence of a typical minimum distance in the given LDPC codes guarantees
that their minimum distance grows linearly with the block length of the code
in an asymptotic sense [25]. It was proved in [26] that a protograph LDPC code
has a typical minimum distance if there is no cycle consisting of only
degree-2 variable nodes in the protograph. Furthermore, in [27], the condition
for a typical minimum distance of the conventional protograph GLDPC codes was
given as follows. Assuming that there are no degree-2 variable nodes with
double edges, i.e., no type 1 degree-2 variable nodes defined in [27], the
neighborhoods of a check node $c_{i}$ that satisfy $d^{i}_{min}\geq
N^{(dv2)}(c_{i})$ are removed along with its edges, where $N^{(dv2)}(c_{i})$
is the number of degree-2 variable nodes among the neighborhoods of $c_{i}$.
This process is repeated until no further degree-2 variable nodes remain.
Then, the GLDPC code has a typical minimum distance if all degree-2 variable
nodes are removed.
The proposed protograph-based PD-GLDPC code also has a similar approach to
that of the conventional protograph GLDPC codes in [27]. However, since a GC
node of the proposed protograph-based PD-GLDPC code is not well defined by a
protograph node, the derivation of the weight enumerator of the proposed
codeword is quite different from that of the conventional GLDPC code. Thus,
the condition for the existence of the typical minimum distance of the
proposed protograph-based PD-GLDPC codes is slightly different from that of
the conventional protograph LDPC code. That is, the degree-2 variable nodes to
be partially doped are initially excluded before the degree-2 variable node
removing process stated in [26], because those variable nodes should be
changed to the variable nodes of degrees higher than two by partial doping.
Thus, we can regard the degree-2 variable nodes to be partially doped as the
variable nodes with higher degrees. Then, we have the following theorem for
the proposed protograph-based PD-GLPDC codes.

######  Theorem 1.

For a $({\mathbb{B}}_{n_{c}\times n_{v}},\mu,\kappa,N,x)$ protograph-based PD-
GLDPC code, if the undoped degree-2 variable nodes in the protograph have no
cycles among themselves, the proposed protograph-based PD-GLDPC code has a
typical minimum distance.

###### Proof.

The proof is given in Appendix A.

The existence of the typical minimum distance of the proposed protograph-based
PD-GLDPC code guarantees that the minimum distance of the proposed code grows
linearly with the code length of the LDPC code, and thus the proposed code has
low error floor for the large code length. In the next section, we use Theorem
1 as the constraint to optimize the protograph in order to guarantee the
existence of the typical minimum distance of the proposed protograph-based PD-
GLDPC code.

###  III-D Comparison between the Proposed PD-GLDPC Code and the Conventional
Protograph GLDPC Code

The main difference between the proposed protograph-based PD-GLDPC codes and
the conventional protograph GLDPC codes is the perspective of doping. While
the conventional protograph GLDPC code replaces an entire row, i.e., a
protograph check node by the parity check matrix of the component code, the
proposed protograph-based PD-GLDPC code appends some rows incident to the
variable nodes copied from single protograph variable node. The focus of the
conventional protograph GLDPC code is to choose a certain protograph check
node, whereas the protograph-based PD-GLDPC code is focusing on choosing which
protograph variable nodes are further protected by partial doping. The
constraint for the conventional protograph GLDPC code is that the check nodes
to be replaced should have the degree equal to the component code length,
while the constraint for the proposed protograph-based PD-GLDPC codes is that
the lifting size of a protograph should be the multiple of the component code
length.
The selection of the check nodes to be replaced in the conventional protograph
GLDPC codes is generally difficult since it requires a combinatory search of
either the best threshold, minimum distance, or convergence speed. Also, since
many SPC nodes of a given base matrix connect the variable nodes with various
degrees, it is very difficult to select the variable nodes to be partially
doped for further protection in the conventional protograph GLDPC codes.
Furthermore, the conventional protograph GLDPC code construction has two
limitations in terms of PEXIT analysis and doping granularity. First, the
PEXIT analysis of the conventional protograph GLDPC codes is not accurate
because the PEXIT of the GC node is derived from averaged EXIT of $\mu$
(component code length) protograph variable nodes. Using a MAP-oriented
computation as in Alg. 1, the GC node outputs a single value that represents
the extrinsic mutual information for the $\mu$ protograph variable nodes.
Thus, a GC node averages up the a priori mutual information of the $\mu$
neighborhood protograph variable nodes and outputs uniform extrinsic mutual
information, which is similar to the EXIT analysis of a random ensemble LDPC
code [21]. Since each GC node receives a priori inputs from $\mu$ different
protograph variable nodes, the EXIT analysis is not accurate. Thus, the higher
variance of the a priori mutual information from the average, the greater the
deviation of the code between the threshold and the actual decoding
performance. On the other hand, since the EXIT of a GC node in the proposed
protograph-based PD-GLDPC code requires the a priori mutual information of the
same protograph variable node, there is no deviation since the $\mu$ mutual
information has the same value, which makes the PEXIT of the proposed code
more accurate.
The second limitation is that the conventional protograph GLDPC codes have
large doping granularity of protograph variable nodes compared to that of the
proposed one. By replacing a single protograph GC node by a component code
with parameters $(\mu,\kappa)$, the $\mu$ protograph variable nodes are doped.
Whereas, for every partial doping of $\beta$ GC nodes in the proposed
protograph-based PD-GLDPC code, the variable nodes copied from single
protograph variable node are partially doped. In other words, the doping
granularity is one, which is smaller than the conventional protograph GLDPC
codes.
In summary, compared to the conventional protograph GLDPC codes, the proposed
protograph-based PD-GLDPC code is more accurate in the PEXIT analysis and has
the smaller doping granularity.

##  IV Constructions of Protograph-based PD-GLDPC Codes

In this section, a protograph-based PD-GLDPC code is constructed from both a
regular protograph and an irregular protograph. Threshold optimization of
protograph-based PD-GLDPC codes is also proposed.

###  IV-A Construction from a Variable Node Regular Protograph

Algorithm 2 The construction method of a protograph-based PD-GLDPC code from a
variable node regular protograph

Input: $w_{r}$, $\mu$, $\kappa$, $n_{v}$, $R$
Output: $y$, $\mathbb{B}_{n_{c}\times n_{v}}$
Step 1) Initialization
For each $y$ in $\lceil\frac{n_{v}-n_{c}+1}{\mu}\rceil\leq
y\leq\lfloor\frac{n_{v}}{\mu}\rfloor$, construct a variable node regular base
matrix $\mathbb{B}$ of size $n_{c}\times n_{v}$ using the PEG method, where
each variable node has degree-$w_{r}$ and $n_{c}=n_{v}(1-R)-(\mu-\kappa)y$.
Step 2) Construction of the candidate protograph-based PD-GLDPC code
For each $y$, partially dope $(\mu,\kappa)$ Hamming codes on $y\mu$ blocks of
variable nodes in the base matrix.
Step 3) Check the existence of the typical minimum distance
For each $y$, $\lceil\frac{n_{v}-n_{c}+1}{\mu}\rceil\leq
y\leq\lfloor\frac{n_{v}}{\mu}\rfloor$, if there exists any cycle for the
submatrix induced by only undoped variable nodes of degree 2, re-initialize
the base matrix as in Step 1) for that $y$.
Step 4) Selection of the optimal protograph-based PD-GLDPC code
For each $y$, compute the PEXIT threshold of the constructed protograph-based
PD-GLDPC code and find $y$ with the best threshold.

We define a variable node regular protograph as a protograph, where all of its
variable nodes have the same degree. Similar to the conventional protograph
GLDPC code construction, we introduce the construction process of a proposed
protograph-based PD-GLDPC code from a regular protograph. In this paper, we
have strictly satisfied the variable node regularity for the protograph while
making the check nodes as regular as possible using the progressive edge-
growth (PEG) algorithm [28]. Since the PEG algorithm is a random construction
method, we have considered doping for variable nodes from the left part of the
base matrix, i.e., doping for $v_{j},\,j=1,2,{\cdots},y\mu$. After lifting by
$N$, the number of partially doped variable nodes is $\mu yN$ and each subset
of size $\mu$ variable nodes lifted from a protograph variable node
constitutes a Hamming code with PCM ${\mathbb{H}}_{Hamm}$ as a component code.
Note that the number of dopings for the protograph variable nodes should be
sufficient enough to satisfy the constraint for the existence of the typical
minimum distance. The necessary condition of $y$ for the typical minimum
distance condition is that the number of undoped degree-2 variable nodes is
less than the number of check nodes of the original protograph, that is,

$n_{v}-\mu y\leq n_{c}-1\leftrightarrow y\geq\frac{n_{v}-n_{c}+1}{\mu}.$

Furthermore, in order to satisfy the sufficient condition for the typical
minimum distance, we make sure that the undoped degree-2 variable nodes do not
form a protograph cycle among themselves. The code rate $R$ of the proposed
protograph-based PD-GLDPC is derived as
$R=1-\frac{n_{c}+(\mu-\kappa)y}{n_{v}}$, where for a given code rate and
$n_{v}$, a protograph-based PD-GLDPC code is constructed. The construction of
a protograph-based PD-GLDPC code from a regular protograph is summarized in
Alg. 2.

###  IV-B Differential Evolution-based Construction from Irregular Protograph

Algorithm 3 Construction of $G_{c}$ and the protograph-based PD-GLDPC code

Input: $\mu$, $\kappa$, $n_{v}$, $R$, $l$, $r$, $y_{max}$
Output: $y^{opt}$, $G_{c}$, $G_{p}$
Step 1) Optimize degree distribution of $G_{c}$
Optimize
$\lambda_{G_{c}}(x)=\lambda_{2}x+\lambda_{3}x^{2}+\lambda_{4}x^{3}+\lambda_{5}x^{4}+\lambda_{6}x^{5}+\lambda_{l}x^{l-1}$
and $\rho_{G_{c}}(x)=\rho_{r-1}x^{r-2}+\rho_{r}x^{r-1}$ using differential
evolution under constraints (a)$\sim$(c): (a) Rate constraint
$R=1-\frac{\int_{0}^{1}\rho_{G_{c}}(x)dx}{\int_{0}^{1}\lambda_{G_{c}}(x)dx},0\leq\lambda_{i}\leq
1,0\leq\rho_{i}\leq 1$, (b) Typical minimum distance constraint
$\frac{\lambda_{2}/2}{\Sigma}\times n_{v}\leq
n_{c}-y_{max}(\mu-\kappa)\leftrightarrow\lambda_{2}\leq\frac{2\Sigma\\{n_{c}-1-y_{max}(\mu-\kappa)\\}}{n_{v}}$,
and (c) $G_{p}$ existence constraint $\lambda_{3}\geq\frac{12\Sigma
y_{max}}{n_{v}},\lambda_{4}\geq\frac{24\Sigma
y_{max}}{n_{v}},\lambda_{5}\geq\frac{20\Sigma
y_{max}}{n_{v}},\lambda_{6}\geq\frac{6\Sigma y_{max}}{n_{v}}$.
Step 2) Construction of $G_{c}$
From the optimized degree distribution and by using the random PEG algorithm,
construct $G_{c}$ defined as $\mathbb{D}^{(2,3,4,5,6,l)}=(a,b,c,d,e,f)$.
Step 3) Optimization of $G_{p}$
For each $y=1,2,{\cdots},y_{max}$, construct $G_{p}$ defined as
$\mathbb{D}^{(2,3,4,5,6,l)}=(a+15y,b-4y,c-6y,d-4y,e-y,f)$ and choose
$y^{opt}\in\\{y\\}$ with the best threshold.
Step 4) Typical minimum distance check of the protograph-based PD-GLDPC code
For the chosen $y^{opt}$ and $G_{p}$, if there exists any degree-2 cycle for
the submatrix induced by undoped variable nodes, go to Step 2). Otherwise,
output $y^{opt}$ and $G_{p}$.

An optimization of an irregular protograph ensemble is made by initially
obtaining the degree distribution of the random ensemble using differential
evolution [29] and constructing the protograph via the PEG algorithm for the
construction of the proposed protograph-based PD-GLPDC codes from irregular
protographs. In this paper, in order to make the check node degrees as even as
possible, we try to construct the protograph from the degree distribution of a
random ensemble. We define $G_{c}$ as the optimized protograph of the
conventional LDPC code and $G_{p}$ as the initial irregular protograph that is
used to construct the protograph-based PD-GLDPC code. $G_{c}$ is constructed
to have the same variable node degree distribution as the protograph-based PD-
GLDPC code constructed from $G_{p}$ after lifting by $N$.
Let $\lambda_{G_{c}}(x)$ and $\rho_{G_{c}}(x)$ be the variable and check node
degree distributions of irregular ensemble to construct $G_{c}$, which is the
optimized protograph for the conventional LDPC codes. In this paper, we assume
the degree distribution of the ensemble of
$\lambda_{G_{c}}(x)=\lambda_{2}x+\lambda_{3}x^{2}+\lambda_{4}x^{3}+\lambda_{5}x^{4}+\lambda_{6}x^{5}+\lambda_{l}x^{l-1}$
and $\rho_{G_{c}}(x)=\rho_{r-1}x^{r-2}+\rho_{r}x^{r-1}$, where $\lambda_{i}$
and $\rho_{i}$ are the portions of edges of variable and check nodes of
degree-$i$. Using the optimized degree distributions of $\lambda_{G_{c}}(x)$
and $\rho_{G_{c}}(x)$, a protograph $G_{c}$ is constructed by the PEG
algorithm. For the description of the protographs that construct the
conventional LDPC codes and the proposed protograph-based PD-GLDPC codes, let
${\mathbb{D}}^{\mathbb{dv}}=(a_{1},{\cdots},a_{max})$ be a
$|\mathbb{dv}|$-sized vector defining the numbers of protograph variable
nodes, where $a_{i}$ is the number of protograph variable nodes of degree
$l_{i}$ and $\mathbb{dv}=\\{l_{1},l_{2},{\cdots},l_{max}\\}$ is a set of
variable node degrees that exist in the protograph. That is, the protograph-
based PD-GLDPC code is constructed from an $n_{c}\times n_{v}$-sized
protograph with ${\mathbb{D}}^{\mathbb{dv}}$ by using the PEG algorithm.
Unlike the regular protograph case, in order to make the same variable node
degree distributions of $G_{c}$ and the protograph-based PD-GLDPC constructed
from $G_{p}$ after lifting by $N$, an optimization of $\lambda_{G_{c}}(x)$ and
$\rho_{G_{c}}(x)$ should be constrained by $y_{max}$, which is the maximum
number of bulks of protograph variable nodes allowed to be partially doped in
$G_{p}$. Then, we make sure that the check node degrees should be as even as
possible for both codes. A protograph-based PD-GLDPC code is constructed by
doping $y\mu$ protograph variable nodes in $G_{p}$. Construction of a
protograph-based PD-GLDPC code from $G_{p}$ is optimized by ranging the doping
bulk $y$, $1\leq y\leq y_{max}$. That is, we search for the optimal value $y$
which maximizes the coding gain between the protograph-based PD-GLDPC codes
constructed from $G_{p}$ and the conventional protograph LDPC codes
constructed from $G_{c}$. Conditions for the degree distributions in order to
construct $G_{c}$ are derived as follows. The conditions need to guarantee two
criteria: i) The variable node degree distributions of the protograph LDPC
code constructed from $G_{c}$ and the protograph-based PD-GLDPC code
constructed from $G_{p}$ after lifting by $N$ are the same and ii) a typical
minimum distance exists for both codes. In this paper, we assume that $\mu
y_{max}$ variable nodes are partially doped only from the degree-2 protograph
variable nodes. For the maximum number of $y_{max}$ bulks of partially doped
protograph variable nodes using the PCM of the $(15,11)$ Hamming code, the
degree distribution of $G_{p}$ should be

$\mathbb{D}^{(2,3,4,5,6,20)}=(a+15y_{max},b-4y_{max},c-6y_{max},d-4y_{max},e-y_{max},f).$

For the existence constraint of the typical minimum distance, the degree
distribution parameters should be non-negative. The parameters $a$$\sim$$f$
are approximated by the PEG construction as

$a\approx\lfloor n_{v}\frac{\lambda_{2}/2}{\Sigma}\rfloor,b\approx\lfloor
n_{v}\frac{\lambda_{3}/3}{\Sigma}\rfloor,c\approx\lfloor
n_{v}\frac{\lambda_{4}/4}{\Sigma}\rfloor,d\approx\lfloor
n_{v}\frac{\lambda_{5}/5}{\Sigma}\rfloor,e\approx\lfloor
n_{v}\frac{\lambda_{6}/6}{\Sigma}\rfloor,\,{\rm{and}}\,f\approx\lfloor
n_{v}\frac{\lambda_{l}/l}{\Sigma}\rfloor,$

where $\Sigma=\int_{0}^{1}\lambda_{G_{c}}(x)dx$. For the realization of the
protograph from the degree distribution using the PEG algorithm, if the
summation $a+b+c+d+e+f$ is lower than $n_{v}$, the values of $a$$\sim$$f$ are
added by 1 in order starting from the lowest variable node degree until the
summation is equal to $n_{v}$.
If $G_{c}$ is determined for a given $y_{max}$ as a variable node degree
distribution of $\mathbb{D}^{(2,3,4,5,6,20)}=(a,b,c,d,e,f)$, where
$a+b+c+d+e+f=n_{v}$, $G_{p}$ defined by
$\mathbb{D}^{(2,3,4,5,6,20)}=(a+15y,b-4y,c-6y,d-4y,e-y,f)$ can be constructed
for $y=1,{\cdots},y_{max}$. By allowing the PEG algorithm of the variable node
degree distribution over a base matrix with size
$\\{n_{c}-(\mu-\kappa)y\\}\times n_{v}$, both the code rate and the variable
node degree distributions for $G_{c}$ and the proposed protograph-based PD-
GLDPC codes constructed from $G_{p}$ after lifting by $N$ are matched. We
search for the value of $y$, which has the best PEXIT threshold while having a
typical minimum distance. The optimized doping value is denoted as $y^{opt}$.
The construction of $G_{c}$ and the PD-GLDPC code is described in Alg. 3.

##  V Numerical Results and Analysis

In this section, we propose the optimized protograph design and show the BLER
of the proposed protograph-based PD-GLDPC codes. The performance of the
conventional protograph LDPC code is compared with that of the proposed
protograph-based PD-GLDPC code. In order to make this comparison, we construct
a protograph ensemble that has the same variable node degree distribution as
that of the proposed protograph-based PD-GLDPC code.

###  V-A Simulation Results for Optimized Protograph-based PD-GLDPC Code from
Regular Protograph

(a)

(b)

Figure 6: Comparison of threshold and BLER for the regular LDPC codes,
irregular protograph LDPC codes, and the protograph-based PD-GLDPC codes from
a regular protograph for code rates $1/2$ and $1/4$.

For numerical analysis, we use a $(15,11)$ systematic Hamming code as the
component code. For a half-rate code, we construct a
$({\mathbb{B}}_{\\{200-4y\\}\times 400},15,11,75,15y)$ protograph-based PD-
GLDPC code, for a given $y$. Since the doping of single protograph variable
node requires $\mu-\kappa=4$ additional protograph rows, the original
protograph should have a total of $200-4y$ rows to construct a half-rate
protograph-based PD-GLDPC code. In order to satisfy the typical minimum
distance property for the proposed PD-GLDPC code, we make sure that there are
no cycles among the undoped degree-2 variable nodes if
$y\geq\frac{400-(200-4y)+1}{15}\leftrightarrow
y\geq\frac{400-200+1}{11}\approx 18.27$ during the PEG process of regular base
matrix construction. For $y=19,20,{\cdots}$, the best threshold is when
$y=19$, having a BEC threshold of 0.444. For comparison, we construct a
$(3,6)$ regular protograph with size $200\times 400$ and lift it by $75$,
which has a threshold of 0.429. Also, by comparing an irregular protograph
LDPC code that has the same degree distribution as the constructed protograph-
based PD-GLDPC code, we can find the performance gain from BP decoding to MAP
decoding over GC nodes. Since the doping of protograph variable nodes in the
proposed protograph-based PD-GLDPC codes occurs only over degree-2 variable
nodes, the degrees of $y\mu$ protograph variable nodes will change. Since the
parity check matrix of a $(15,11)$ systematic Hamming code has 4 degree-1
columns, 6 degree-2 columns, 4 degree-3 columns, and 1 degree-4 column, the
conventional irregular LDPC protograph without doping having the same variable
node degree distribution with the proposed protograph-based PD-GLDPC code is
defined as ${\mathbb{D}}^{(2,3,4,5,6)}=(115,76,114,76,19)$. Irregular
protograph LDPC code and the (3,6) regular protograph LDPC code are
constructed from the $200\times 400$ protograph and the protograph-based PD-
GLDPC code is constructed from the $124\times 200$ protograph with
$x=y\mu=285$. The irregular protograph LDPC code also has the same variable
node degree distribution as the proposed protograph-based PD-GLDPC code, where
it is defined by $\mathbb{D}^{(2,3,4,5,6)}=(115,76,114,76,19)$. Using the PEG
algorithm, we construct the irregular protograph LDPC code, having a threshold
of 0.4234. The BLER performances of the constructed irregular protograph LDPC
code, protograph-based PD-GLDPC code, and a $(3,6)$ regular code with
$(n,k)=(30000,15000)$ are shown in Fig. 6(a), where all three codes are half
rate LDPC codes constructed by $N=75$. The coding gain of the constructed
protograph-based PD-GLDPC code is 0.021 and 0.015 compared to irregular
protograph LDPC code and $(3,6)$ regular protograph LDPC code, respectively.
Similarly, we construct a $({\mathbb{B}}_{\\{300-4y\\}\times
400},15,11,105,15y)$ protograph-based PD-GLDPC code for code rate $1/4$. For
$y\geq\frac{400-(300-4y)+1}{15}\leftrightarrow
y\geq\frac{400-300+1}{11}\approx 9.18$, the optimized threshold is the highest
value $0.6935$ at $y=10$. Similar to the half-rate case, we construct a
$(3,4)$ regular protograph LDPC code using the PEG algorithm to have a
threshold of $0.647$. The irregular protograph ensemble has a variable degree
distribution of ${\mathbb{D}}^{(2,3,4,5,6)}=(250,40,60,40,10)$ and the
threshold is $0.6883$. The irregular protograph LDPC code and the $(3,4)$
regular protograph LDPC code are constructed from the $300\times 400$
protograph and the protograph-based PD-GLDPC code is constructed from the
$260\times 400$ protograph with $x=y\mu=150$. The irregular protograph also
has the same variable node degree distribution as the protograph-based PD-
GLDPC code, where it is defined by
$\mathbb{D}^{(2,3,4,5,6)}=(250,40,60,40,10)$. All the codes are lifted by
$105$ to make $(42000,10500)$ codes and the numerical results are in Fig.
6(b). The coding gain of the constructed protograph-based PD-GLDPC code is
0.0052 and 0.0465 compared to irregular protograph LDPC code and $(3,4)$
regular protograph LDPC code, respectively.

###  V-B Simulation Result for Optimized Protograph-based PD-GLDPC Code from
Irregular Protograph

Figure 7:  Comparison of threshold and BLER for the irregular protograph LDPC
code from $G_{c}$, the conventional random ensemble GLDPC code from [17], and
the protograph-based PD-GLDPC code from $G_{p}$ for code rate $1/2$. TABLE I:
Simulation results for optimized protograph-based PD-GLDPC code from irregular
protographs

$y_{max}$ |  $\lambda_{G_{c}}(x),\rho_{G_{c}}(x)$ (threshold) |  |  $G_{c}$ protograph
---
$\mathbb{D}^{(2,3,4,5,6,20)}$
/ $G_{c}$ threshold
|  $G_{p}$ protograph
---
$\mathbb{D}^{(2,3,4,5,20)}$, $y^{opt}$
/ PD-GLDPC threshold
Coding gain
5 |  | $\lambda_{G_{c}}(x)=0.2049x+0.2489x^{2}+0.1150x^{3}$
---
$+0.074x^{4}+0.0210x^{5}+0.3363x^{19}$
$\rho_{G_{c}}(x)=0.9735x^{7}+0.0265x^{8}$ ($0.4815$)
| $(165,134,47,23,5,26)$
---
/ $0.4620$
|  $(240,114,17,3,26)$, $y^{opt}=5$
---
/ $0.4699$
$0.0079$
10 |  | $\lambda_{G_{c}}(x)=0.1894x+0.2255x^{2}+0.1431x^{3}$
---
$+0.1191x^{4}+0.0357x^{5}+0.2872x^{19}$
$\rho_{G_{c}}(x)=0.9908x^{7}+0.0012x^{8}$ ($0.4696$)
| $(152,121,57,38,9,23)$
---
/ $0.4523$
|  $(287,85,3,2,23)$, $y^{opt}=9$
---
/ $0.4638$
$0.0115$
15 |  | $\lambda_{G_{c}}(x)=0.1632x+0.1758x^{2}+0.2143x^{3}$
---
$+0.1827x^{4}+0.0543x^{5}+0.2098x^{19}$
$\rho_{G_{c}}(x)=0.9940x^{7}+0.0060x^{8}$ ($0.4476$)
| $(131,94,86,59,14,16)$
---
/ $0.4352$
|  $(341,38,2,3,16)$, $y^{opt}=14$
---
/ $0.4534$
$0.0182$

The construction for the protograph of the conventional protograph LDPC code,
$G_{c}$ is made for $y_{max}=5,10,15$ for the half-rate protograph ensemble.
The numerical results are summarized in Table I, where the coding gain given
for the proposed PD-GLDPC code is compared to the conventional protograph LDPC
code with equal degree distribution. The random ensemble-based GLDPC code with
threshold $0.466$ in [17] is represented as a variable node ensemble
$\lambda(x)=0.8x^{2}+0.01x^{3}+0.01x^{5}+0.18x^{7}$ and a doping portion of
$40\%$ by Hamming code. Fig. 7 shows the performance comparison of three half-
rate codes which are the irregular protograph LDPC code constructed from
$G_{c}$, a random ensemble-based GLDPC code in [17], and the proposed
protograph-based PD-GLDPC code constructed from $G_{p}$, where $y_{max}=5$.
All three codes in Fig. 7 are $(n,k)=(30000,15000)$ codes of half-rate, where
$G_{c}$ is defined as $\mathbb{D}^{(2,3,4,5,6,20)}=(165,134,47,23,5,26)$ and
has the same variable node degree distribution as the PD-GLDPC code after
lifting by $N=105$. $x=y\mu=75$ protograph variable nodes are partially doped
in the PD-GLDPC code. The constructed protograph-based PD-GLDPC code for
$y_{max}=5$ has a coding gain of 0.0079 and 0.0039 compared to the GLDPC code
in [17] and the irregular protograph LDPC from $G_{c}$, respectively. Fig. 7
shows that the proposed protograph-based PD-GLDPC code has a good performance
both in the waterfall and low error floor region due to the fact that the code
is optimized by increasing the doping as much as possible, and at the same
time, the typical minimum distance constraint is satisfied. In terms of the
asymptotic analysis, increasing the portion of degree-2 variable nodes
increases the possibility of the code to approach the channel capacity [24].
However, the existence of a typical minimum distance of the protograph is also
important, which upper bounds the portion of degree-2 variable nodes in the
LDPC code. Thus, balancing of degree-2 variable nodes is needed in order to
satisfy both a typical minimum distance condition and a good threshold. The
proposed protograph-based PD-GLDPC code guarantees the balancing of the
degree-2 variable nodes by carefully choosing the rate of the protograph code
and the number of doping on degree-2 variable nodes.

Figure 8:  Comparison of threshold and BLER for the irregular protograph LDPC
code from $G_{c}$ and the proposed protograph-based PD-GLDPC code from $G_{p}$
for code rate $1/4$.

Similarly, the construction of $G_{c}$ is made for the code rate $1/4$ case.
We simulate the proposed protograph-based PD-GLDPC with the small doping case,
where the optimization of random ensemble for $G_{c}$ is done under the
constraint $y_{max}=3$. The degree distribution of the optimized results is
obtained as

$\lambda_{G_{c}}(x)=0.2858x+0.1218x^{2}+0.0533x^{3}+0.0369x^{4}+0.0095x^{5}+0.4927x^{29},$
$\rho_{G_{c}}(x)=0.9995x^{5}+0.0005x^{6},$

which makes $G_{c}$ with $\mathbb{D}^{(2,3,4,5,6,30)}=(258,74,24,13,2,29)$ via
the PEG algorithm. The threshold of the proposed protograph-based PD-GLDPC
code is best when $y=2$ and for this instance, $G_{p}$ is then defined as
$\mathbb{D}^{(2,3,4,5,30)}=(288,66,12,5,29)$. Both codes are
$(n,k)=(42000,10500)$ codes of code rate $1/4$, where $G_{c}$ is defined as
$\mathbb{D}^{(2,3,4,5,6,30)}=(258,74,24,13,2,29)$ and has the same variable
node degree distribution as $G_{p}$ after lifting by $N=105$. $x=y\mu=30$
protograph variable nodes are partially doped in the protograph-based PD-GLDPC
code. The simulation result of both the irregular protograph LDPC code and the
proposed protograph-based PD-GLDPC code are shown for the same variable node
degree distribution in Fig. 8. The thresholds of $G_{c}$ and the proposed PD-
GLDPC code are 0.6702 and 0.7031, respectively, where the coding gain is
0.0329 compared to the irregular protograph LDPC code.

##  VI Conclusion

We proposed a new construction method of a protograph-based PD-GLDPC code that
has advantages the accuracy of the PEXIT analysis and a finer granularity of
the protograph node doping compared to the conventional protograph GLDPC
codes. Also, the proposed optimized PD-GLDPC codes constructed from regular
and irregular protographs have good BLER performances compared to the regular
protograph LDPC codes and conventional GLDPC code constructed from irregular
random ensemble. While the PEXIT of a conventional GLDPC code over a GC node
averages out the a priori and extrinsic mutual information, the proposed
protograph-based PD-GLDPC code does not need such an averaging process because
the mutual information exchanges over the same protograph variable node.
Furthermore, since it is possible to partially dope the protograph variable
nodes with a granularity 1, the design of a protograph-based PD-GLDPC code
becomes more flexible in terms of balancing the rate loss and doping.

##  Appendix A Proof of Theorem 1: The Constraint for the Existence of the
Typical Minimum Distance of the Proposed Protograph-based PD-GLDPC Codes

A proof for the constraint of the existence of a typical minimum distance for
the proposed protograph-based PD-GLDPC codes is given in this appendix.
Similar to that in [27], a typical minimum distance is driven by the weight
enumerator analysis over the lifted protograph. In order to use the notations
in [27], we’ve distinguished the indexing notations during the enumeration for
the partially doped variable nodes using ′. Also, the $c_{j}$ and $v_{i}$
notations are used for the check nodes and the variable nodes, respectively.
Suppose that the proposed protograph-based PD-GLDPC code is constructed from
the protograph defined by $G=(V,C,E)$ and the $x$ variable nodes are partially
doped, where component codes are identical with the parameters $(\mu,\kappa)$.
We are given a variable node set $V=\\{v_{1},{\cdots},v_{n_{v}}\\}$ and a
check node set $C_{PD{\text{-}}GLDPC}=B_{Hamm}\cup
C=\\{b_{1},{\cdots},b_{x}\\}\cup\\{c_{1},{\cdots},c_{n_{c}}\\}$ for the
protograph. It is important to note that the GC node set $B_{Hamm}$ is not
defined over a protograph. However, the codeword enumeration can be made when
the protograph is lifted, where $b_{i^{\prime}},i^{\prime}\in[x]$ is a virtual
check node that represents the Hamming check nodes used for partial doping for
$v_{i^{\prime}}$ in the original protograph. Although $b_{i^{\prime}}$ is not
a protograph check node, we define it for the enumeration of the partially
doped protograph variable nodes. The protograph-based PD-GLDPC code is
constructed by lifting the graph $G$ by $N$ times and permuting the replicated
edges. Each $v_{i}$ ($c_{j}$) has degree $q_{v_{i}}$ ($q_{c_{j}}$) and each
$b_{i^{\prime}}$ has degree $\mu$. For the enumeration of the GC node
$b_{i^{\prime}}$, we can think of it as a protograph node of degree $\mu$ that
is lifted by a factor of $\frac{N}{\mu}$. The upper bound of the weight
enumerator of the proposed protograph-based PD-GLDPC code with weight $d$,
denoted as $A_{d}^{PD{\text{-}}GLDPC}$ is derived as follows.
Let $w_{m,u},u\in[q_{v_{m}}]$ be the $u$th edge weight from a variable node
$v_{m}$. For a partially doped variable node $v_{m},m\in[x]$, there are $\mu$
weights sent towards the incident GC node, where the $u$th weight is defined
as $w^{\prime}_{m,u},u\in[\mu]$. For a given input weight vector
$\mathbb{d}=(d_{1},{\cdots},d_{n_{v}})$, we need to calculate
$A^{PD{\text{-}}GLDPC}(\mathbb{d})$ and sum it over every instance of
$\mathbb{d}$ that satisfies $d=d_{1}+{\cdots}+d_{n_{v}}$. For input
$d_{i^{\prime}},i^{\prime}\in[x]$, it is clear that
$\sum_{i=1}^{\mu}w^{\prime}_{m,i}=d_{i^{\prime}}$ because the extrinsic weight
$w^{\prime}_{m,i}$ consists of weights solely from $v_{m}$. We introduce the
following notations:

  * •

$A^{v_{i}}_{d_{i}}(\mathbb{w}_{i})=\binom{N}{d_{i}}\delta_{d_{i},w_{i,1}},{\cdots},\delta_{d_{i},w_{i,q_{v_{i}}}}=\begin{cases}\binom{N}{d_{i}},\quad\mbox{if}\,w_{i,j}=d_{i},\forall
j\in[q_{v_{i}}]\\\ 0,\quad\mbox{otherwise}\end{cases}$ is the vector weight
enumerator for a variable node $v_{i}$ of the protograph [27].

  * •

$A^{c_{j}}(\mathbb{z}_{j})$ is the vector weight enumerator for a check node
$c_{j}$ of the original protograph, for the incoming weight vector
$\mathbb{z}_{j}=[z_{j,1},{\cdots},z_{j,q_{c_{j}}}]$ [27].

  * •

$B_{d_{i^{\prime}}}^{v_{i^{\prime}}}(\mathbb{w}^{\prime}_{i^{\prime}})=\begin{cases}1,\quad\mbox{for}\,w^{\prime}_{i^{\prime},1}+{\cdots}+w^{\prime}_{i^{\prime},\mu}=d_{i^{\prime}}\\\
0,\quad\mbox{otherwise}\end{cases}$ is the vector weight enumerator for
partially doped variable nodes $v_{i^{\prime}},\,i^{\prime}\in[x]$.

  * •

$B^{b_{i^{\prime}}}(\mathbb{w}^{\prime}_{i^{\prime}})$ is the vector weight
enumerator for check nodes that are created during the lifting process given
the weight vector $\mathbb{w}^{\prime}_{i^{\prime}}$.
$A^{b_{i^{\prime}}}(d_{i^{\prime}})$ is the summation of enumerators over all
possible $\mathbb{w}^{\prime}_{i^{\prime}}$ values given that
$w^{\prime}_{i^{\prime},1}+{\cdots}+w^{\prime}_{i^{\prime},\mu}=d_{i^{\prime}}$
satisfying

$\displaystyle
A^{b_{i^{\prime}}}(d_{i^{\prime}})=\sum_{\mathbb{w}^{\prime}}B^{b_{i^{\prime}}}(\mathbb{w}^{\prime}_{i^{\prime}})=\sum_{\mathbb{w}^{\prime}}\sum_{\\{\mathbb{m}\\}}C(\frac{N}{\mu};m_{1},{\cdots},m_{K}),$

where $\mathbb{w}^{\prime}=(w_{1}^{\prime},{\cdots},w_{\mu}^{\prime})$ such
that
$\sum_{i=1}^{\mu}w_{1}^{\prime}=d_{k}^{\prime},\,w_{i}^{\prime}\leq\frac{N}{\mu}.$

Then, the weight enumerator is given as

$\displaystyle A_{d}^{PD-GLDPC}=\sum_{\\{\mathbb{d}\\}}A^{PD-
GLDPC}(\mathbb{d}),$

where

$\displaystyle A^{PD{\text{-}}GLDPC}(\mathbb{d})$
$\displaystyle=\frac{\prod_{i=1}^{n_{v}}A^{v_{i}}_{d_{i}}(\mathbb{w}_{i})\prod_{j=1}^{n_{c}}A^{c_{j}}(\mathbb{z}_{j})\times\prod_{i^{\prime}=1}^{x}B_{d_{i^{\prime}}}^{v_{i^{\prime}}}(\mathbb{w}^{\prime}_{i^{\prime}})B^{b_{i^{\prime}}}(\mathbb{w}^{\prime}_{i^{\prime}})}{\prod_{s=1}^{n_{v}}\prod_{r=1}^{q_{v_{s}}}\binom{N}{w_{s,r}}\times\prod_{s^{\prime}=1}^{x}\prod_{r^{\prime}=1}^{\mu}\binom{\frac{N}{\mu}}{w^{\prime}_{s^{\prime},r^{\prime}}}}$
$\displaystyle=\sum_{\\{\mathbb{w}^{\prime}_{i^{\prime}}:\,w^{\prime}_{i^{\prime},1}+{\cdots}+w^{\prime}_{i^{\prime},\mu}=d_{i^{\prime}}\\}}\frac{\prod_{j=1}^{n_{c}}A^{c_{j}}(\mathbb{d}_{j})\times\prod_{i^{\prime}=1}^{x}B^{b_{i^{\prime}}}(\mathbb{w}^{\prime}_{i^{\prime}})}{\prod_{i=1}^{n_{v}}\binom{N}{d_{i}}^{q_{v_{i}}-1}\times\prod_{s^{\prime}=1}^{x}\prod_{r^{\prime}=1}^{\mu}\binom{\frac{N}{\mu}}{w^{\prime}_{s^{\prime},r^{\prime}}}}.$

The solution to the equation $\mathbb{w}^{\prime}=\mathbb{m}\mathbb{M^{C}}$ is
given as $\mathbb{m}=\\{m_{1},{\cdots},m_{K}\\}$. The term
$\binom{\frac{N}{\mu}}{w^{\prime}_{s^{\prime},r^{\prime}}}$ is lower bounded
by
$(\frac{N}{\mu})^{w^{\prime}_{s^{\prime},r^{\prime}}}e^{-w^{\prime}_{s^{\prime},r^{\prime}}\cdot{\rm{ln}}\,w^{\prime}_{s^{\prime},r^{\prime}}}$.
Then, $A^{PD-GLDPC}(\mathbb{d})$ can be upper bounded as

$\displaystyle A^{PD{\text{-}}GLDPC}(\mathbb{d})$
$\displaystyle\leq\sum_{\\{\mathbb{w}^{\prime}_{i^{\prime}}:\,w^{\prime}_{i^{\prime},1}+{\cdots}+w^{\prime}_{i^{\prime},\mu}=d_{i^{\prime}}\\}}\frac{\prod_{j=1}^{n_{c}}A^{c_{j}}(\mathbb{d}_{j})\times\prod_{i^{\prime}=1}^{x}B^{b_{i^{\prime}}}(\mathbb{w}^{\prime}_{i^{\prime}})}{\prod_{i=1}^{n_{v}}\binom{N}{d_{i}}^{q_{v_{i}}-1}\times\prod_{s^{\prime}=1}^{x}\prod_{r^{\prime}=1}^{\mu}(\frac{N}{\mu})^{w^{\prime}_{s^{\prime},r^{\prime}}}e^{-w^{\prime}_{s^{\prime},r^{\prime}}\cdot{\rm{ln}}\,w^{\prime}_{s^{\prime},r^{\prime}}}}$
$\displaystyle\leq\sum_{\\{\mathbb{w}^{\prime}_{i^{\prime}}:\,w^{\prime}_{i^{\prime},1}+{\cdots}+w^{\prime}_{i^{\prime},\mu}=d_{i^{\prime}}\\}}\frac{\prod_{j=1}^{n_{c}}A^{c_{j}}(\mathbb{d}_{j})\times\prod_{i^{\prime}=1}^{x}B^{b_{i^{\prime}}}(\mathbb{w}^{\prime}_{i^{\prime}})}{\prod_{i=1}^{n_{v}}\binom{N}{d_{i}}^{q_{v_{i}}-1}\times\prod_{s^{\prime}=1}^{x}(\frac{N}{\mu})^{d_{s^{\prime}}}e^{-d_{s^{\prime}}\cdot{\rm{ln}}\,d_{s^{\prime}}}}$
$\displaystyle\leq\sum_{\\{\mathbb{w}^{\prime}_{i^{\prime}}:\,w^{\prime}_{i^{\prime},1}+{\cdots}+w^{\prime}_{i^{\prime},\mu}=d_{i^{\prime}}\\}}\frac{\prod_{j=1}^{n_{c}}A^{c_{j}}(\mathbb{d}_{j})\times\prod_{i^{\prime}=1}^{x}B^{b_{i^{\prime}}}(\mathbb{w}^{\prime}_{i^{\prime}})}{\prod_{i=1}^{n_{v}}\binom{N}{d_{i}}^{q_{v_{i}}-1}\times(\frac{N}{\mu})^{P}e^{-P\cdot{\rm{ln}}\,P}}$
$\displaystyle=\frac{\prod_{j=1}^{n_{c}}A^{c_{j}}(\mathbb{d}_{j})\times\prod_{i^{\prime}=1}^{x}\sum_{\\{\mathbb{w}^{\prime}_{i^{\prime}}:\,w^{\prime}_{i^{\prime},1}+{\cdots}+w^{\prime}_{i^{\prime},\mu}=d_{i^{\prime}}\\}}B^{b_{i^{\prime}}}(\mathbb{w}^{\prime}_{i^{\prime}})}{\prod_{i=1}^{n_{v}}\binom{N}{d_{i}}^{q_{v_{i}}-1}\times(\frac{N}{\mu})^{P}e^{-P\cdot{\rm{ln}}\,P}}$
$\displaystyle=\frac{\prod_{j=1}^{n_{c}}A^{c_{j}}(\mathbb{d}_{j})\times\prod_{i^{\prime}=1}^{x}A^{b_{i^{\prime}}}(d_{i^{\prime}})}{\prod_{i=1}^{n_{v}}\binom{N}{d_{i}}^{q_{v_{i}}-1}\times(\frac{N}{\mu})^{P}e^{-P\cdot{\rm{ln}}\,P}},$
(3)

where $P=\sum_{s^{\prime}=1}^{x}d_{s^{\prime}}$ is the total weight of the $x$
partially doped variable nodes. Then
$\sum_{t}(t\cdot{\rm{ln}}\,t)\leq(\sum_{t}t)\cdot{\rm{ln}}\,(\sum_{t}t)$ is
used for the second and the third inequalities in (3). It was shown in (18) of
[27] that the inequality

$\frac{\prod_{j=1}^{n_{c}}A^{c_{j}}(\mathbb{d}_{j})}{\prod_{i=1}^{n_{v}}\binom{N}{d_{i}}^{q_{v_{i}}-1}}\leq\prod_{i=1}^{n_{v}}e^{(q_{v_{i}}-1-\frac{q_{v_{i}}}{d_{min}^{(c)}})d_{i}{\rm{ln}}\,\frac{d_{i}}{N}+\frac{q_{v_{i}}(2+k_{max}^{(c)}{\rm{ln}\,2)}}{d_{min}^{(c)}}d_{i}}$

holds, where $d_{min}^{(c)}$ is the minimum distance of an SPC component code
for the original protograph and $k_{max}^{(c)}$ is the maximum number of
codewords of an SPC component code. Using the similar notations in [27], let
$d_{min}^{(b)}$ and $k^{(b)}$ be the minimum distance and the number of
codewords of the $(\mu,\kappa)$ component code for the GC nodes. Then,
$\prod_{i^{\prime}=1}^{x}A^{b_{i^{\prime}}}(d_{i^{\prime}})$ is upper bounded
as

$\displaystyle\prod_{i^{\prime}=1}^{x}A^{b_{i^{\prime}}}(d_{i^{\prime}})$
$\displaystyle\leq\prod_{i^{\prime}=1}^{x}\sum_{\\{\mathbb{w}^{\prime}_{i^{\prime}}:\,w^{\prime}_{i^{\prime},1}+{\cdots}+w^{\prime}_{i^{\prime},\mu}=d_{i^{\prime}}\\}}\prod_{i=1}^{\mu}{(\frac{N}{\mu})}^{\frac{1}{d_{min}^{(b)}}w^{\prime}_{i^{\prime},i}}e^{\frac{(2+k^{\prime}_{i^{\prime}}ln\,2)}{d_{min}^{(b)}}w^{\prime}_{i^{\prime},i}-\frac{1}{d_{min}^{(b)}}w^{\prime}_{i^{\prime},i}{\rm{ln}}\,w^{\prime}_{i^{\prime},i}}$
$\displaystyle=\prod_{i^{\prime}=1}^{x}\sum_{\\{\mathbb{w}^{\prime}_{i^{\prime}}:\,w^{\prime}_{i^{\prime},1}+{\cdots}+w^{\prime}_{i^{\prime},\mu}=d_{i^{\prime}}\\}}(\frac{N}{\mu})^{\frac{1}{d_{min}^{(b)}}d_{i^{\prime}}}e^{\frac{(2+k^{\prime}_{i^{\prime}}{\rm{ln}}\,2)}{d_{min}^{(b)}}d_{i^{\prime}}-\sum_{i=1}^{\mu}\frac{1}{d_{min}^{(b)}}w^{\prime}_{i^{\prime},i}{\rm{ln}}\,w^{\prime}_{i^{\prime},i}}$
$\displaystyle\leq\prod_{i^{\prime}=1}^{x}\sum_{\\{\mathbb{w}^{\prime}_{i^{\prime}}:\,w^{\prime}_{i^{\prime},1}+{\cdots}+w^{\prime}_{i^{\prime},\mu}=d_{i^{\prime}}\\}}(\frac{N}{\mu})^{\frac{1}{d_{min}^{(b)}}d_{i^{\prime}}}e^{\frac{(2+k^{\prime}_{i^{\prime}}{\rm{ln}}\,2)}{d_{min}^{(b)}}d_{i^{\prime}}-\frac{1}{d_{min}^{(b)}}d_{i^{\prime}}{\rm{ln}}\,\frac{d_{i^{\prime}}}{\mu}}$
$\displaystyle\leq\prod_{i^{\prime}=1}^{x}\binom{d_{i^{\prime}}+\mu-1}{d_{i^{\prime}}}(\frac{N}{\mu})^{\frac{1}{d_{min}^{(b)}}d_{i^{\prime}}}e^{\frac{(2+k^{\prime}_{i^{\prime}}{\rm{ln}}\,2)}{d_{min}^{(b)}}d_{i^{\prime}}-\frac{1}{d_{min}^{(b)}}d_{i^{\prime}}{\rm{ln}}\,\frac{d_{i^{\prime}}}{\mu}}.$
(4)

For the inequality in the third line of (4), we use the fact that
$\sum_{i=1}^{p}t_{i}{\rm{ln}}\,t_{i}\leq s\cdot{\rm{ln}}\,\frac{s}{p}$ with
$s=t_{1}+{\cdots}+t_{p}$, which is clear by using the derivative on the
multivariable function that consists of independent $t_{i}$’s. The equality is
satisfied when all $t_{i}$ values are the same. Going back to (3), let
$f(P)=\frac{1}{\binom{N}{\mu}^{P}e^{-P{\rm{ln}}\,P}}$ for convenience. Then we
have

$\displaystyle A^{PD{\text{-}}GLDPC}(\mathbb{d})\leq$
$\displaystyle\prod_{i=1}^{n_{v}}e^{(q_{v_{i}}-1-\frac{q_{v_{i}}}{d_{min}^{(c)}})d_{i}{\rm{ln}}\,\frac{d_{i}}{N}+\frac{q_{v_{i}}(2+k_{max}^{(c)}{\rm{ln}\,2)}}{d_{min}^{(c)}}d_{i}}$
$\displaystyle\times\prod_{i^{\prime}=1}^{x}e^{d_{i^{\prime}}+\mu-1}(\frac{N}{\mu})^{\frac{1}{d_{min}^{(b)}}d_{i^{\prime}}}e^{\frac{(2+k^{\prime}_{i^{\prime}}{\rm{ln}}\,2)}{d_{min}^{(b)}}d_{i^{\prime}}-\frac{1}{d_{min}^{(b)}}d_{i^{\prime}}{\rm{ln}}\,\frac{d_{i^{\prime}}}{\mu}}\times
f(P)$ $\displaystyle\leq$
$\displaystyle\prod_{i=1}^{n_{v}}e^{(q_{v_{i}}-1-\frac{q_{v_{i}}}{d_{min}^{(c)}})d_{i}{\rm{ln}}\,\frac{d_{i}}{N}+\frac{q_{v_{i}}(2+k_{max}^{(c)}{\rm{ln}\,2)}}{d_{min}^{(c)}}d_{i}}$
$\displaystyle\times
e^{x(\mu-1)}e^{P}\prod_{i^{\prime}=1}^{x}(\frac{N}{\mu})^{\frac{1}{d_{min}^{(b)}}d_{i^{\prime}}}e^{\frac{(2+k^{(b)}{\rm{ln}}\,2)}{d_{min}^{(b)}}d_{i^{\prime}}-\frac{1}{d_{min}^{(b)}}d_{i^{\prime}}{\rm{ln}}\,\frac{d_{i^{\prime}}}{\mu}}\times
f(P).$ (5)

We classify the variable nodes in the protograph into three groups before
doping:

  * •

Protograph variable nodes of degrees higher than 2

  * •

Protograph variable nodes of degree 2 to be partially doped

  * •

Protograph other variable nodes of degree 2.

We also separate the weights of codewords after lifting into three parts
according to the three groups of variable nodes: $u_{i}$, $p_{z}$, and
$l_{j}$, where $u_{i}$ is the weight of the sub-codeword corresponding to a
protograph variable node $v_{i}$ of degree higher than 2 and $p_{z}$ and
$l_{j}$ are the weights of codewords of each partially doped and undoped
protograph variable node $v_{z}$ and $v_{j}$ of degree 2 from the protograph,
respectively. The sum of sub-codeword weights for each group of variable nodes
is given as $U=\sum_{i}u_{i}$, $P=\sum_{z}p_{z}$, and $L=\sum_{j}l_{j}$. It is
clear that for the total codeword weight $d$, $d=U+P+L$. Then, the upper bound
of the first term in (5) is written as

$\displaystyle\prod_{i=1}^{n_{v}}e^{(q_{v_{i}}-1-\frac{q_{v_{i}}}{d_{min}^{(c)}})d_{i}{\rm{ln}}\,\frac{d_{i}}{N}+\frac{q_{v_{i}}(2+k_{max}^{(c)}{\rm{ln}\,2)}}{d_{min}^{(c)}}d_{i}}\leq$
$\displaystyle
e^{(2-\frac{3}{d_{min}^{(c)}})(d-P-L){\rm{ln}}\,\frac{d-P-L}{N}+\frac{3(2+k_{max}^{(c)})}{d_{min}^{(c)}}\cdot(d-P-L)}$
$\displaystyle\times e^{\frac{2(2+k_{max}^{(c)})}{d_{min}^{(c)}}\cdot L}\cdot
e^{\frac{2(2+k_{max}^{(c)})}{d_{min}^{(c)}}\cdot P},$ (6)

which is derived by using three weight groups of codewords similar to (20) of
[27]. We share the same inequality
$u_{i}<Ne^{-\frac{(2+k_{max}^{(c)}{\rm{ln}}\,2)}{d_{min}^{(c)}-1}}$ over the
given codeword weight $d$ as in [27]. Using the derivation in [27], the upper
bound of the second term of (5) can be derived as

$\displaystyle\prod_{i^{\prime}=1}^{x}(\frac{N}{\mu})^{\frac{1}{d_{min}^{(b)}}d_{i^{\prime}}}e^{\frac{(2+k^{(b)}{\rm{ln}}\,2)}{d_{min}^{(b)}}d_{i^{\prime}}-\frac{1}{d_{min}^{(b)}}d_{i^{\prime}}{\rm{ln}}\,\frac{d_{i^{\prime}}}{\mu}}=\prod_{i^{\prime}=1}^{x}e^{\frac{1}{d_{min}^{(b)}}d_{i^{\prime}}{\rm{ln}}\,\frac{N}{\mu}}e^{\frac{(2+k^{(b)}{\rm{ln}}\,2)}{d_{min}^{(b)}}d_{i^{\prime}}-\frac{1}{d_{min}^{(b)}}d_{i^{\prime}}{\rm{ln}}\,\frac{d_{i^{\prime}}}{\mu}}$
$\displaystyle=\prod_{i^{\prime}=1}^{x}e^{\frac{1}{d_{min}^{(b)}}d_{i^{\prime}}{\rm{ln}}\,\frac{N}{d_{i^{\prime}}}}e^{\frac{(2+k^{(b)}{\rm{ln}}\,2)}{d_{min}^{(b)}}d_{i^{\prime}}}=\prod_{z=1}^{x}e^{\frac{1}{d_{min}^{(b)}}p_{z}{\rm{ln}}\,\frac{N}{p_{z}}}e^{\frac{(2+k^{(b)}{\rm{ln}}\,2)}{d_{min}^{(b)}}p_{z}}$
$\displaystyle\leq
e^{\frac{1}{d_{min}^{(b)}}P\cdot{\rm{ln}}\,\frac{Nx}{P}}e^{\frac{(2+k^{(b)}{\rm{ln}}\,2)}{d_{min}^{(b)}}P}.$
(7)

Using (6) and (7), the upper bound of $A^{PD{\text{-}}GLDPC}(\mathbb{d})$ is
derived in terms of $E(d,P,L)$ as follows:

$\displaystyle A^{PD{\text{-}}GLDPC}(\mathbb{d})\leq$ $\displaystyle
e^{\frac{1}{d_{min}^{(b)}}P\cdot{\rm{ln}}\,\frac{Nx}{P}}e^{\frac{(2+k^{(b)}{\rm{ln}}\,2)}{d_{min}^{(b)}}P}\cdot
e^{(2-\frac{3}{d_{min}^{(c)}})(d-P-L){\rm{ln}}\,\frac{d-P-L}{N}+\frac{3(2+k_{max}^{(c)})}{d_{min}^{(c)}}\cdot(d-P-L)}$
$\displaystyle\times
e^{\frac{2(2+k_{max}^{(c)})}{d_{min}^{(c)}}\cdot(P+L)}\cdot
e^{x(\mu-1)}e^{P}\cdot f(P).$ (8)

Let $E(d,P,L)$ be the parameter satisfying
$A^{PD{\text{-}}GLDPC}(\mathbb{d})\leq e^{x(\mu-1)}\cdot e^{E(d,P,L)}$. Then,
from the upper bound in (8), $E(d,P,L)$ is given as

$\displaystyle E(d,P,L)=$
$\displaystyle\frac{1}{d_{min}^{(b)}}P\cdot{\rm{ln}}\,\frac{Nx}{P}+\frac{(2+k^{(b)}{\rm{ln}}\,2)}{d_{min}^{(b)}}P+(2-\frac{3}{d_{min}^{(c)}})(d-P-L){\rm{ln}}\,\frac{d-P-L}{N}$
$\displaystyle+\frac{3(2+k_{max}^{(c)})}{d_{min}^{(c)}}\cdot(d-P-L)+\frac{2(2+k_{max}^{(c)})}{d_{min}^{(c)}}\cdot(P+L)+P+P{\rm{ln}}\,P-P{\rm{ln}}\,\frac{N}{\mu}.$

Assuming that there are no type 1 degree-2 variable nodes defined in [27] and
there are no cycles consisting only of undoped variable nodes of degree-2, we
use the result of (22) in [27] such that the inequality
$l_{2,k}^{(c_{j})}\leq\frac{1}{d_{min}^{(c_{j})}}(L_{2}^{(c_{j})}+\sum_{i}w_{i}^{(c_{j})})$
is satisfied for all $j\in[n_{c}]$, where $l_{2,k}^{(c_{j})}$ is the weight of
the degree-2 undoped variable node of $G_{p}$ and the total weight of them is
denoted as $L_{2}^{(c_{j})}$ for check node $c_{j}$. Similar to the result in
[27], we can derive the upper bound $L\leq\gamma(U+P)$, which is the same as
$L\leq\frac{\gamma}{1+\gamma}d$. Now, the upper bound of $E(d,P,L)$ needs to
be derived for independent values $L$ and $P$. The first and second partial
derivatives of $E(d,P,L)$ by $P$ are given as

$\displaystyle\frac{dE}{dP}=$
$\displaystyle\frac{1}{d_{min}^{(b)}}\cdot{\rm{ln}}\,\frac{Nx}{eP}+\frac{(2+k^{(b)}{\rm{ln}}\,2)}{d_{min}^{(b)}}-(2-\frac{3}{d_{min}^{(c)}}){\rm{ln}}\,\frac{e(d-P-L)}{N}-\frac{(2+k_{max}^{(c)}{\rm{ln}}\,2)}{d_{min}^{(c)}}$
$\displaystyle+1+{\rm{ln}}\,P+1-{\rm{ln}}\,\frac{N}{\mu}<0,$
$\displaystyle\frac{d^{2}E}{dP^{2}}=$
$\displaystyle-\frac{1}{d_{min}^{(b)}P}+(2-\frac{3}{d_{min}^{(c)}})\cdot\frac{1}{d-P-L}+\frac{1}{P}>0.$

Since the first derivative over $P$ is negative and the second derivative is
positive, $E(d,P,L)$ is upper bounded by

$\underset{P\to
0+}{\lim}E(d,P,L)=(2-\frac{3}{d_{min}^{(c)}})(d-L){\rm{ln}}\,\frac{d-L}{N}+\frac{3(2+k_{max}^{(c)})}{d_{min}^{(c)}}\cdot(d-L)+\frac{2(2+k_{max}^{(c)}{\rm{ln}}\,2)}{d_{min}^{(c)}}L.$

Since the resulting upper bound of $E(d,L)$ is the same as (37) in [27], the
rest of the proof is the same as that in [27] and thus the proposed constraint
guarantees the existence of typical minimum distance of the proposed
protograph-based PD-GLDPC code.

## Acknowledgment

The authors would like to thank…

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