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
0
=1,α, ···, α
q−2
, give all elements of
GF(q). For any nonzero element δ = α
i
with 0 ≤ i<q− 1,
we form a (q −1)-tuple over GF(q), z(δ) =(z
0
, z
1
, ···, z
q−2
),
where the ith component z
i
= 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
i
(x) of an element α
i
is the
smallest degree polynomial that has α
i
as a root. The minimal
polynomial φ
i
(x) has binary coefficients and is irreducible
over GF(2)= {0,1} [17]. Moreover, φ
i
(x) has roots α
i
, α
2i
,
..., α
2
κ−1
i
, where κ divides m. These elements are known as
the conjugate elements of α
i
in GF(2
m
), and they can build a
conjugate coset M
i
,
M
i
= {α
i
,α
2
i, ··· ,α
2
κ−1
i
}. (2)
The exponents of the elements in M
i
form a cyclotomic coset
(CC) [17],
C
i
= {i, 2i, ··· , 2
κ−1
i}. (3)
CCs partition the set I
2
m
−1
of integers modulo 2
m
−1, namely,
C
i
∩ C
j
= ∅, for i = j, and ∪
i
C
i
= I
2
m
−1
,
___________________________________
978-1-4673-2197-6/12/$31.00 ©2012 IEEE
1468
IEEE 2012 International conference on Signal Processing, Beijing, Oct.21-25, 2012
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