1
Design of regular (2, d
c
)-LDPC codes over
GF(q) using their binary images
Charly Poulliat
∗†
, Marc Fossorier
†
, David Declercq
∗
∗
ETIS, ENSEA/UCP/CNRS
6 Avenue du Ponceau
F-95014 Cergy-Pontoise, France.
Email: {charly.poulliat, david.declercq}@ensea.fr
†
Dept. of Electrical Eng.
University of Hawaii at Manoa
2540 Dole St., Honolulu, HI-96822, USA.
Email: marc@spectra.eng.hawaii.edu
Abstract
In this paper, a method to design regular (2, d
c
)-LDPC codes over GF(q) with both good waterfall
and error floor properties is presented, based on the algebraic properties of their binary image. First, the
algebraic properties of rows of the parity check matrix H associated with a code are characterized and
optimized to improve the waterfall. Then the algebraic properties of cycles and stopping sets associated
with the underlying Tanner graph are studied and linked to the global binary minimum distance of the
code. Finally, simulations are presented to illustrate the excellent performance of the designed codes.
Index Terms
channel coding, error correction coding, nonbinary LDPC codes, iterative decoding, binary image.
This work has been partially supported by the Newcom UE Network of Excellence
March 20, 2007 DRAFT
2
I. INTRODUCTION
Since their rediscovery in [16], low density parity check (LDPC) codes designed over GF(q)
have been shown to approach the Shannon limit performance for q = 2 and very long code lengths
[15][22]. Some efficient optimization methods of the code profile and the matrix structure have
been derived for both long [22][3] and moderate [13] length cases. For fields with parameters
q > 2, it has been shown that the error performance can be improved for moderate code lengths
by increasing q [5][4][11]. It has been shown, especially in [5][11], that as q becomes large
(q ≥ 64) the best performances at finite length are obtained for “ultra-sparse” LDPC codes,
that is with the minimum connectivity on the symbol nodes d
v
= 2. Furthermore, it is shown
in [11] that d
v
= 2 non binary LDPC codes have optimal average Hamming weight spectrum
as q → +∞ and N → +∞ when used on binary input channels. In this paper, we will focus
on the finite length optimization of d
v
= 2 non binary LDPC codes, for which the problem of
choosing appropriately the non zero values in the parity check matrix is simplified. Note also
that the decoding complexity of codes in GF(q) is a lot larger than for binary codes, but iterative
decoding of non binary LDPC codes using the belief propagation (BP) algorithm or its simplified
versions has been addressed efficiently by several authors [5][1][6].
The design of non binary LDPC codes can be addressed in order to meet different objectives:
(i) performance, by trying to improve the waterfall region and/or to lower the error floor, and (ii)
decoding complexity versus performance tradeoff, by trying to ensure good overall performance
using only a limited set of parameters for some efficient and practical hardware implementation
purposes. For finite length codes, the optimization problem is generally solved in a disjoint
manner. First, the positions of the nonzero entries of the parity check matrix H associated with
the non binary code are optimized in order to have good girth properties and minimize the impact
of cycles, when using the BP algorithm on the associated Tanner graph. This can be efficiently
done using the progressive edge growth (PEG) algorithm [13]. Then, the nonzero entries can be
selected either randomly from a uniform distribution among nonzero elements of GF(q) [13] or
carefully to meet some design criteria as done in [4][17].
In this paper, we address the problem of the selection and the matching of the parity check
matrix nonzero entries assuming that the positions of nonzero entries in the parity check matrix
H associated with the non binary code have been previously optimized. The proposed method
DRAFT March 20, 2007
3
is based on the binary image representation of the matrix H and of its components. First we
address the problem of rows optimization as previously done in [5][17] in order to improve
the waterfall region. Then, we address the problem of lowering the error floor: based on the
observation that the columns defining the minimum distance in the binary image of H are
located on symbols belonging to the shortest length cycles and the associated stopping sets, we
propose a method intended to improve the minimum distance of the binary image of the code.
To this end, we use the algebraic properties of both cycles and stopping sets, considered as
topological substructures inherently present in the underlying Tanner graph of the code. Finally,
the complexity-performance tradeoff is addressed: we show for example that for regular (2, 4)
and (2, 8) LDPC codes, using only one optimized row of coefficients to generate the parity check
matrix, it is possible to have at least the same performance as for a code with randomly selected
coefficients and, for some fields, the waterfall and the error floor region can be both improved.
The paper is organized as follows: in Section II, we briefly review the binary image con-
struction of a non binary parity check matrix and the vector representation of the parity check
equations. The optimization of the rows of the parity check matrix is addressed for waterfall
improvement in Section III. We also study the thresholds under density evolution for random
and row optimized code ensembles. Section IV provides a study of the binary representation of
both cycles and stopping sets, and establishes links between those topological structures of the
Tanner graph and the binary minimum distance property of the code. This study allows us to
propose a method to improve the error floor when using the row optimized code ensemble. In
Section V, some optimization and simulation results are provided and finally conclusions and
perspectives are drawn in Section VI.
II. BINARY IMAGES OF A NON BINARY PARITY CHECK MATRIX H
The motivation of using the binary image of the LDPC code is essentially that we address and
illustrate the optimization process for the non zero values in the case of binary input additive
white gaussian noise (BI-AWGN). In this context, our goal is then to try to maximize the
Hamming minimum distance at the bit level of the LDPC code. Note however that using the
binary image of the code is not mandatory and one could easily generalize our approach at the
symbol level, as it will be notified in sections IV-B, IV-C and IV-E.
Let us consider the parity check matrix H associated with a regular non binary LDPC code
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with the parameters (d
v
, d
c
, N) representing the number of nonzero entries of H for the columns,
for the rows and the code length respectively. All the nonzero elements of H are elements of
the Galois field GF(q), with q = 2
p
and q is the order of the field. Nonzero elements belong to
the set S =
α
k
: k = 0 . . . q − 2
where α is the primitive element of the field.
A. Representation of the Galois field using matrices
The Galois field GF(q), described usually using a polynomial (or vector) representation, can
be also represented using matrices [18, p.106]
Definition 1: If p(x) = a
0
+ a
1
x + . . . + x
p
is a polynomial of degree p having its coefficients
in GF(2). The companion matrix of p(x) is the p × p matrix
A =
0 1 0 . . . 0
0 0 1 . . . 0
0 0 0 . . . 1
a
0
a
1
a
2
. . . a
p−1
The characteristic polynomial of this matrix is given by
det(A − xI) = p(x)
where I is the identity matrix.
If p(x) is a primitive polynomial, it can be shown [18] that the matrix A is the primitive
element of the Galois field GF (2
p
) under a matrix representation and thus the powers of A are
the nonzero elements of this field, defining the set M =
0, A
k
: k = 0 . . . q − 2
. Additions
and multiplications in the field correspond to additions and multiplications modulo 2 of these
matrices.
B. Vector representation for the parity check equations
Based on the matrix representation of each nonzero entry, we give thereafter the equivalent
vector representation of the parity check equations associated with the rows of H.
Let x = [x
0
. . . x
N−1
] be a codeword. For the i−th parity equation of H, we have
X
j:h
ij
6=0
h
ij
x
j
= 0 (1)
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5
Translating (1) into the vector domain, we can write
X
j:h
ij
6=0
H
ij
x
j
t
= 0
t
where H
ij
is the transpose of the matrix representation of the Galois field element h
ij
, x
j
is the
vector representation (binary mapping) of the symbol element x
j
and t holds for transpose. The
vector 0 is the all zero component vector.
Considering the i-th parity check equation of H, we define H
i
= [H
ij
0
. . . H
ij
m
. . . H
ij
d
c
−1
]
as the equivalent binary parity check matrix, with {j
m
: m = 0 . . . d
c
− 1} the indexes of the
nonzero elements of the i−th row. Let X
i
= [x
j
0
. . . x
j
d
c
−1
] be the binary representation of the
symbols of the codeword x involved in the i−th parity check equation. When using the binary
representation, the i-th parity check equation of H, can be written as
H
i
X
i
t
= 0
t
We define d
min
(i) as the minimum distance of the binary code associated with H
i
.
C. Example
Let p(x) = x
3
+ x + 1 be the primitive polynomial used to generate the elements of GF (2
3
).
The primitive element for the matrix representation is given by
A =
0 1 0
0 0 1
1 1 0
Thus, {A
k
: k = 0, . . . , 6} are the nonzero elements of GF (2
3
) under this matrix representation
and it is readily checked for our example that A
k
t
α
l
t
= α
k+l
t
.
III. SELECTING ROWS FOR WATERFALL IMPROVEMENT
In this section, we investigate the choice of “good” rows for the parity check matrix regardless
of the structure of the Tanner graph associated with it. First, we briefly review the method
proposed in [5], [17] to select the coefficients row by row. We show that the set of rows provided
by [17] can easily be reduced and we give an analysis of these coefficient sets using the binary
images of the code considered. Then, since the method of [5], as an instance of density evolution,
can be computationally expensive for high field orders, we propose a simpler optimization method
March 20, 2007 DRAFT
![TABLE I ROW COEFFICIENTS FORGF(16) AND GF(64) AND dc = 4 FROM [17].](/figures/table-i-row-coefficients-forgf-16-and-gf-64-and-dc-4-from-17-9eo8y1f5.webp)
![Fig. 6. FER versusEb/N0 : comparison with [8]. Non binary codes are designed over GF(256). K = 1024 information bits](/figures/fig-6-fer-versuseb-n0-comparison-with-8-non-binary-codes-are-2u3wqrzr.webp)
