294 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 10, NO. 2, MARCH/APRIL 2004
Block-Circulant Low-Density Parity-Check Codes
for Optical Communication Systems
Olgica Milenkovic, Member, IEEE, Ivan B. Djordjevic, and Bane Vasic, Senior Member, IEEE
Abstract—A novel family of structured low-density parity-check
(LDPC) codes with block-circulant parity-check matrices that con-
sist of permutation blocks is proposed. The codes from this family
are based on new combinatorial objects termed cycle-invariant dif-
ference sets, and they have low storage requirements, fast encoding
algorithms, and girth of at least six. Most importantly, they tend to
outperform many other known structured LDPCs of comparable
rate and length.
Index Terms—Cycle-invariant difference sets, forward error
correction, low-density parity-check codes, optical communica-
tions.
I. INTRODUCTION
T
HE role of forward error correction (FEC) in maintaining
acceptable transmission quality is becoming increasingly
important as the capacity of dense wavelength-division multi-
plexing (DWDM) long-haul transmission systems grows. FEC
can improve the performance of optical communications sys-
tems without introducing major changes to optical materials and
devices and without major paradigm shifts in hardware such as
converting to all-optical networks. This characteristic is espe-
cially important as the physical properties of optical networks
are being pushed to their limits. The FEC proposed in this paper
can readily be adapted to follow any shift in system implemen-
tation.
In state-of-the-art optical communication systems, two FEC
standards are currently in use, both based on Bose–Chaud-
huri–Hocquenghen (BCH) codes [1]. For example, “in-band”
FEC used in SONET/SDH framing is a 3-[4359, 4320] short-
ened BCH code (ITU-T G.707 standard). “Outband” FEC
coding is accomplished by using interleaved Reed–Solomon
(RS) [255, 238] codes, as specified in the digital wrapper of
the ITU G.709 standard. The FEC used in submarine systems
is based on a Reed–Solomon code with parameters [255, 239],
as specified by the ITU-T G.975 standard [2]. For optical
communication systems with bit rates exceeding 10 Gb/s, more
powerful FEC codes with large coding gains and high code
rates are desirable. Two such schemes have been proposed
Manuscript received September 2, 2003; revised January 18, 2004. This work
was supported by the National Science Foundation under Grant ITR 0325979.
O. Milenkovic is with the Department of Electrical and Computer
Engineering, University of Colorado, Boulder, CO 80303 USA (e-mail:
milenkov@colorado.edu).
I. B. Djordjevic is with the Department of Electrical and Computer Engi-
neering, University of Arizona, Tucson, AZ 85719 USA and the Faculty of
Computing, Engineering and Mathematical Sciences, University of the West of
England, Bristol, U.K. (e-mail: ivan@ece.arizona.edu; icadj@yahoo.com).
B. Vasic is with the Department of Electrical and Computer Engineering, Uni-
versity of Arizona, Tucson, AZ 85719 USA (e-mail: vasic@ece.arizona.edu).
Digital Object Identifier 10.1109/JSTQE.2004.827832
recently: Ait Sab et al. [3] proposed a concatenated scheme
with two RS codes, and Pyndiah [4] and Ait Sab et al. [5]
proposed block turbo codes.
Recently, the authors showed in [6]–[8] that the error-cor-
recting capability of turbo codes can be surpassed by low-den-
sity parity-check (LDPC) codes; at the same time, the hardware
complexity of LDPC codes tends to be significantly smaller than
that corresponding to turbo codes [9]. LDPC codes are usually
designed in a pseudorandom fashion [10], but this leads to en-
coders/decoders with complexity exceeding practical limits im-
posed on high-speed optical communication systems. A more
attractive approach that we propose in this paper is to design
block-circulant LDPC codes with permutation blocks based on
a new class of combinatorial objects; these structures allow for
small storage requirements, as well as for performing encoding
and decoding via fast and simple circuits. Such high-speed FEC
architectures are of crucial importance in optical communica-
tions, and there has been a great deal of activity in this area. For
example, Agere Systems (see Azadet et al. [11]) reported an op-
tical networking interface device with four parallel RS codecs,
each operating at 2.5 Gb/s. Additionally, Agere Systems [12],
[13] and Flarion [14] researchers developed codecs employing
LDPC designs that seem to be a combination of structured and
random-like components.
The main result of this paper is a novel class of LDPC
codes based on a combinatorial structure referred to as an
-fold cycle-invariant difference set (CIDS). The parity-check
matrices of the proposed codes are block-circulants with
permutation blocks. Both the minimum distance and the girth
of the corresponding bipartite graph [8] are at least six. By
carefully designing the cycle-invariant difference set, and by
appropriately permuting the elements in some block-rows,
code graphs with
( an odd prime) variable
nodes can be made to have girth at least 2
2. As the authors
demonstrated by extensive computer simulations in [8], codes
with high girth tend to perform very well, since a larger number
of message-passing iterations can be performed before extrinsic
information of nodes in the graph becomes correlated. Due
to the block-circulant structure of the parity-check matrix,
encoding can be performed very efficiently. Additionally,
the min-sum decoding algorithm employed for soft-decision
decoding of these codes requires a number of binary logic
operations at least one order of magnitude smaller than that
corresponding to hard-decision decoding of RS or turbo codes
[4], [5] with comparable performance. For more details on
min-sum algorithm complexity, the reader is referred to [17,
Appendix B] and [20]–[22].
1077-260X/04$20.00 © 2004 IEEE
Fec在光纤通信系统中的应用
ldpc码在光纤通信系统中的应用
本篇文章中的ldpc码的设计特色
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