短长度不规则非二进制QC-LDPC码：环群陪集构造法

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"本文提出了一种基于克隆陪集法构造不规则非二进制准循环低密度奇偶校验(QC-LDPC)码的新方法，尤其适用于短码长的情况。利用这种方法构建的QC-LDPC码在迭代解码下，在加性高斯白噪声(AWGN)信道中的误比特性能、错误地板和解码收敛性方面表现出色，同时保持较低的解码复杂度。关键词包括克隆陪集、非二进制LDPC码、准循环和低解码复杂度。" 不规则非二进制准循环低密度奇偶校验(QC-LDPC)码是一种重要的纠错编码技术，它们属于LDPC码的范畴，而LDPC码最早由Gallager在1962年提出，作为一种逼近香农极限的编码方式。相比于二进制LDPC码，非二进制LDPC码通常能在较短的码长下实现更优秀的性能，这主要得益于非二进制域的更大符号空间，能提供更好的纠错能力。在本文中，作者Meng Lu和Lijun Zhang提出了一种新的构造不规则非二进制QC-LDPC码的方法——克隆陪集法。该方法利用了伽罗华域上的克隆陪集理论，这是一种在编码领域中常用的设计工具，特别是对于构造准循环码。克隆陪集法允许构造出具有特定特性的码字，如非规则度分布，这在提升码性能方面至关重要。不规则性是LDPC码的一个关键特性，它指的是码字中检查节点和变量节点的连接度（即邻接度）不是均匀分布的。不规则的结构可以优化迭代解码过程，从而提高码的性能。通过克隆陪集法构建的不规则非二进制QC-LDPC码在AWGN信道中表现出良好的性能，这意味着它们在对抗噪声干扰时有较高的抗错误能力。此外，文章强调了这些码在迭代解码中的错误地板和解码收敛性的优良表现。错误地板是指在低信噪比(SNR)区域，解码错误率出现的突然升高现象，一个好的编码设计应尽量降低错误地板。解码收敛性则关乎解码算法在多轮迭代后达到稳定状态的速度，快速的收敛性意味着更低的计算资源需求。这项研究提供了一种有效且高效的构造短码长不规则非二进制QC-LDPC码的方法，对于实际应用，尤其是需要在有限资源下实现高性能通信系统的场景，如无线通信、数据存储和卫星通信等领域，具有很高的价值。其低解码复杂度和良好性能指标表明，这种码设计方法有望成为未来编码理论和技术发展的重要参考。

Constructions of Irregular Nonbinary QC-LDPC

Codes: Cyclotomic Coset Approach

Meng Lu and Lijun Zhang

School of Electronic and Information Engineering

Beijing Jiaotong University

Beijing, 100044, China

Email: ljzhang@bjtu.edu.cn

Abstract—A cyclotomic coset approach for constructions of

irregular nonbinary quasi-cyclic low-density parity-check (QC-

LDPC) codes of short length is proposed. QC-LDPC codes

obtained by this method perform reasonably well over the

additive white Gaussian noise (AWGN) channel with iterative

decoding in terms of bit-error performance, error-ﬂoor, and

decoding convergence with low decoding complexity.

Index Terms—cyclotomic coset; nonbinary LDPC codes; quasi-

cyclic; low decoding complexity

I. INTRODUCTION

Low-density parity-check (LDPC) codes as a Shannon limit

approaching class [1–4] can be constructed in both binary

and high-order Galois ﬁelds. But in general, binary LDPC

codes need very long code length to approach the Shannon

limit [5]. Nonbinary LDPC codes, ﬁrst investigated by Davey

and MacKay in 1998 [6, 7], have shown performance advan-

tages over their binary counterpart with relatively short length,

but at the expense of higher decoding complexity [6]. After

then, many efforts have been made to reduce the decoding

computational complexity for practical application and this

have become a research focus recently [8–11].

As regards nonbinary LDPC codes, the construction meth-

ods can be classiﬁed into two major categories: (i) random

nonbinary codes constructed by computer using certain criteria

or rules, such as MacKay codes [6] and PEG codes [12]; and

(ii) structured nonbinary codes constructed based on algebraic

theory, which usually have good algebraic structure. Addi-

tive subgroups, cyclic subgroups, parallel ﬂats, superposition-

dispersion, and perfect different set are often used to construct

nonbinary codes based on algebraic tools [13–16]. However,

there exist two problems with respect to the algebraic con-

struction methods mentioned above. First, theses codes exhibit

good performance, but are relatively long, which inevitably

results in high implementation complexity. Second, although,

in general, irregular LDPC codes outperform the regular one,

most of recent methods only pay attention to regular codes,

and few efforts have been made on irregular nonbinary LDPC

codes.

In this paper, we propose a cyclotomic coset approach

to obtain irregular and powerful LDPC codes with short

length. Using different combinations of cyclotomic cosets

while building base matrix, a class of irregular nonbinary QC-

LDPC codes can be obtained which perform comparatively

well. It should be noted although the CC approach tries to

build a sub-space with the help of the CC, it is totally different

from Lin’s method in [14], which is based on the ﬁnite ﬁeld.

II. α-M

ULTIPLIED CPM AND CYCLOTOMIC COSET

A. α-Multiplied CPM

Let α be a primitive element of GF(q). Then, the powers

of α, α

−∞

=0,α

=1,α, ···, α

q−2

, give all elements of

GF(q). For any nonzero element δ = α

with 0 ≤ i<q− 1,

we form a (q −1)-tuple over GF(q), z(δ) =(z

, z

, ···, z

q−2

where the ith component z

= δ and all other components are

equal to zero. Create a (q − 1) × (q − 1) matrix A over GF(q)

with z(δ), z(αδ), ···, z(α

q−2

δ) as its rows,

A =

⎡

⎢

⎣

z(δ)

z(αδ)

z(α

q−2

δ)

⎤

⎥

⎦

. (1)

Matrix A can be regarded as a special form of circulant

where each row is the right cyclic-shift of the row above it

multiplied by α and the ﬁrst row is the right cyclic-shift of

the last row multiplied by α. We call A the q-ary α-multiplied

circulant permutation matrix (CPM) of the nonzero element δ

in GF( q) [17].

B. Cyclotomic Coset

The minimal polynomial φ

(x) of an element α

is the

smallest degree polynomial that has α

as a root. The minimal

polynomial φ

(x) has binary coefﬁcients and is irreducible

over GF(2)= {0,1} [17]. Moreover, φ

(x) has roots α

, α

..., α

κ−1

, where κ divides m. These elements are known as

the conjugate elements of α

in GF(2

), and they can build a

conjugate coset M

= {α

,α

i, ··· ,α

κ−1

}. (2)

The exponents of the elements in M

form a cyclotomic coset

(CC) [17],

= {i, 2i, ··· , 2

κ−1

i}. (3)

CCs partition the set I

−1

of integers modulo 2

−1, namely,

∩ C

= ∅, for i = j, and ∪

= I

−1

___________________________________

1468

IEEE 2012 International conference on Signal Processing, Beijing, Oct.21-25, 2012

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