构造长度12的Type-1 QC-LDPC码：性能优良的中等速率选择
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本文主要探讨了Type-1 QC-LDPC（Quasi-Cyclic Low-Density Parity-Check）码的显式构造，这是一种特殊的低密度奇偶校验码，其在编码理论中有重要应用。作者Guohua Zhang和Rudolf Mathar，IEEE高级会员，针对给定的J×J（J> 3）阶矩阵，其中任意两个行向量之间的差最多只包含Z_P（有限域上的整数集）中的一个元素，提出了一个编码方法。这种限制确保了代码的循环特性，即它们具有周期性结构。
编码构造的关键在于使用（3，L）-正规准循环（QC）结构，其中3<L≤J。编码长度被设定为P×L^2，而编码的周长（girth）达到12，这意味着最少的循环路径长度为12，有助于提高纠错能力和避免短周期错误的影响。在低中等码率和长度下，新构造的Type-1 QC-LDPC码表现出良好的性能。
在编码技术方面，与传统的只由同大小的循环排列矩阵（CPM，Circulant Permutation Matrix）构成的QC-LDPC码相比，Type-1 QC-LDPC码引入了零矩阵（ZM），这提供了更多的灵活性和设计选项。零矩阵的存在允许对码的结构进行更精细的调整，从而优化编码效率和性能。
文章的贡献在于提供了一种新的编码构造方法，它不仅扩展了现有的QC-LDPC码类型，还展示了在特定参数范围内如何实现高效的编码。为了验证这些构造的有效性，文中包含了相应的仿真结果，显示了新构造的Type-1 QC-LDPC码在实际通信系统中的优越性能，特别是在码率适中且码长较长的情况下。
本文的核心知识点包括：Type-1 QC-LDPC码的定义、构造原理、关键参数（如长度、周长、（3，L）-正规性）、以及与传统QC-LDPC码的区别，以及通过实验证明的新码在特定应用场景下的优势。这对于理解低密度奇偶校验码的设计和优化，特别是在无线通信和其他需要高效纠错能力的领域，具有重要的学术价值和实践意义。
460 IEEE COMMUNICATIONS LETTERS, VOL. 21, NO. 3, MARCH 2017
Explicit Constructions for Type-1 QC-LDPC Codes With Girth 12
Guohua Zhang and Rudolf Mathar, Senior Member, IEEE
Abstract—Given any J × J ( J > 3) square matrix over
Z
P
such that the differences of any two row vectors contain
each element in Z
P
at most once, a class of (3, L)-regular quasi-
cyclic low-density parity-check codes is explicitly constructed
with lengths PJL
2
and girth 12, where L is any integer satisfying
3 < L ≤ J. Simulation results show that the new codes perform
very well for moderate rates and lengths.
Index Terms— Low-density parity-check (LDPC) codes,
quasi-cyclic (QC), circulant permutation matrix (CPM), girth.
I. INTRODUCTION
A
low-density parity-check (LDPC) code is defined as
the null space of its sparse parity-check matrix (PCM).
A classical QC-LDPC code [1] is associated with a PCM
consisting of circulant permutation matrices (CPMs) of the
same size P, while a type-1 QC-LDPC code is associated with
a PCM composed of both CPMs and zero matrices (ZMs) of
the same size P. Compared with classical QC-LDPC codes,
their type-1 counterparts usually enable a larger upper bound
in terms of minimum distance [2], which in turn hopefully
leads to a better decoding performance.
If a PCM has R 1’s in each column and L 1’s in each row,
then the associated LDPC code is called (R, L)-regular.
Girth is the length of the shortest cycles in the
Tanner graph corresponding to a PCM. To the best of our
knowledge, Tanner’s method [3] and Jing’s one [4] are the only
existing ways to explicitly construct (3, L)-regular structured
QC-LDPC codes with girth 12. However, Tanner’s method
only works for classical codes with L = 5andaprimeP,
while Jing’s one only for type-1 codes with a prime P.
In this Letter, we present a novel method to construct type-1
(3, L)-regular QC-LDPC codes with girth exactly 12 for
any P, which naturally includes Jing’s method as a special
case.
Our method is based on a type of J × J matrix over
Z
P
in which the differences of any two rows do not contain
repetitive elements. Obviously, such matrices can be directly
used to construct (J, J )-regular QC-LDPC codes free of
4-cycles (if the fact that the rate is most often zero is not
considered). From such matrices, column-weight-two LDPC
codes with girth 12 can be constructed by the approach
Manuscript received October 29, 2016; revised November 20, 2016;
accepted November 22, 2016. Date of publication December 1, 2016; date
of current version March 8, 2017. This work was supported by the China
Scholarship Council (CSC) and the National Natural Science Foundation of
China under Grant 61471294. The associate editor coordinating the review of
this letter and approving it for publication was Z. Ma.
G. Zhang is with the Institute for Theoretical Information Technology,
RWTH Aachen University, 52074 Aachen, Germany, and also with the
China Academy of Space Technology (Xi’an), Xi’an 710100, China (e-mail:
zhangghcast@163.com).
R. Mathar is with the Institute for Theoretical Information Technology,
RWTH Aachen University, 52074 Aachen, Germany (e-mail: mathar@ti.
rwth-aachen.de).
Digital Object Identifier 10.1109/LCOMM.2016.2634022
in [5], while our method enables column-weight-three type-1
QC-LDPC codes with girth 12.
II. C
ONSTRUCTION
The MP× NP PCM H for a classical or type-1 QC-LDPC
code with length NP can be determined by its correspond-
ing M × N exponent matrix E and the CPM size P [6].
Precisely, when H is generated from E, the element ∞ within
E corresponds to a P × P ZM and any other element (say e)
corresponds to a P × P identity matrix with rows cyclically
shifted to the right by e (mod P) positions. Denote by g(E, P)
the girth of the Tanner graph associated with an exponent
matrix E and the CPM/ZM size P.
A. A Class of Exponent Matrices
In this Letter, we investigate a special type of exponent
matrices E which can be viewed as an array of matrices as
follows:
E =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
S ∞
0
··· ∞
0
∞
0
S ··· ∞
0
.
.
.
.
.
.
.
.
.
.
.
.
∞
0
∞
0
··· S
W ∞
0
··· ∞
0
∞
0
W ··· ∞
0
.
.
.
.
.
.
.
.
.
.
.
.
∞
0
∞
0
··· W
C
0
C
1
··· C
J −1
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
, (1)
where S, W and ∞
0
are all J × J
2
matrices, and C
u
(0 ≤
u ≤ J − 1) is a J
2
× J
2
matrix. Therefore, E is a
3J
2
×J
3
matrix. To be specific, (i) S is a J ×J arraydefinedby
S =
⎡
⎢
⎢
⎢
⎣
S
0
∞
1
··· ∞
1
∞
1
S
1
··· ∞
1
.
.
.
.
.
.
.
.
.
.
.
.
∞
1
∞
1
··· S
J −1
⎤
⎥
⎥
⎥
⎦
, (2)
where ∞
1
is a 1 × J matrix of ∞’s, and S
j
is a 1 × J
matrix denoted by [s
j,0
s
j,1
··· s
j, J −1
],for0≤ j ≤ J − 1;
(ii) W is a 1× J array of W
0
’s, i.e. W =[W
0
, W
0
, ··· , W
0
],
where W
0
is a J × J matrix with 0’s on the main diagonal
and ∞’s otherwise; (iii) ∞
0
is a J × J
2
matrix composed of
∞’s; (iv) C
u
(0 ≤ u ≤ J − 1) is a J × J array defined by
⎡
⎢
⎢
⎢
⎣
D(T
0
(S
u
)) ∞
2
··· ∞
2
∞
2
D(T
1
(S
u
)) ··· ∞
2
.
.
.
.
.
.
.
.
.
.
.
.
∞
2
∞
2
··· D(T
J −1
(S
u
))
⎤
⎥
⎥
⎥
⎦
. (3)
For any vector x =[x
0
, x
1
, ··· , x
J −1
], T
k
(x) denotes x with
elements cyclically shifted to the left by k (mod J) positions,
1558-2558 © 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
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