
Research Article


An Overview of Low Density Parity Check Codes 

Abid Yahya,
Farid Ghani,
R. Badlishah
and
Raj Malook



ABSTRACT






Received: March 15, 2010;
Accepted: May 22, 2010;
Published: July 10, 2010


INTRODUCTION
Low Density Parity Check (LDPC) codes (Gallager, 1963),
have acquired considerable attention due to its nearcapacity error execution
and powerful channel coding technique, with an adequately long codeword length.
The performance of LDPC codes is investigated, at many events of interests and
is encountered to outperform turbo codes with good error correction (Berrou
et al., 1993; MacKay, 1999; Chung
et al., 2001).
The proof of the minimum distance of the code, for most existing classes of
algebraic codes makes important use of the algebraic structure of the code (Tanner
et al., 2001). The minimum distance for each class of codes is usually
recognized by using a proof method exact to that precise class, and the minimum
distance of codes in a new class may not be easily accomplished (MacKay,
1999; Tanner et al., 2001). Tanner
et al. (2001) presented a class of codes called Sparse Difference
Codes, in which the parity check matrix of the resultant linear code is a block
structure as a permutation matrix. The socalled array codes have paritycheck
matrices with powers of permutation matrices, and can be considered as LDPC
codes (Fan, 2000). Eleftheriou and
Olcer (2002) proposed this class of LDPC codes for an application in Digital
Subscriber Lines (DSL) that employ discrete multitone modulation. Authors emphasize
on further investigation in order to completely utilize the LDPC coding for
DSLs, together with very high DSL, and to evaluate performance with actual loop
and noise characteristics. The simulation results show that, even under rigid
latency constraints, good net coding gains can be achieved by LDPC coding. Moreover,
LDPC codes do not exhibit error floors at the low biterror rates of interest
for DSL transmission. Furthermore, Kim et al. (2002)
proposed explicit construction of LDPC codes based on partial row construction
with girth at least eight.
Authors propose construction of LDPC codes based on partial row construction and also the Sparness of codes are not well defined. Moreover, authors observe no error floor but at low SNR, while error floor occurs at high SNR. This leads to design codes which exhibit no error floors at high SNR. Simulation results show that the proposed method improves the code rate only at low girth codes.
Algebraically structured QCLDPC codes: The construction of LDPC codes
from circulant permutation matrices is investigated by Fossorier
(2004). It is observed from simulation results that such codes cannot have
a Tanner graph representation with larger girth and their minimum distance cannot
be increased by increasing the code length and as a result, the girth as for
random constructions. Fossorier derives a simple necessary and sufficient condition
for the Tanner graph (Tanner, 1981) of the QuasiCyclic
(QC) LDPC codes to have a given girth. It is shown from the simulation results
that the proposed codes have a girth g of at most 12, which generalizes the
result simulated by Tanner et al. (2001). In
case of girth g = 6, the condition is very easy to construct QCLDPC codes,
which perform significantly better at moderate block lengths ,when iteratively
decoded with the Belief Propagation (BP) algorithm (MacKay,
1999). Actually, Fossorier employs an upper bound on the minimum Hamming
distance of QC LDPC codes and necessary condition to reach this bound. Fossorier
has found that an appropriate coset weight distribution (MacKay
and Postol, 2003) has to be considered when constructing families of LDPC
codes. Another family of structured LDPC codes with girth six is introduced
by Milenkovic and Laendner (2004) based on a class of
idempotent, symmetric Latin and modified Latin squares. The proposed codes have
a block structure with permutation block, which ascertains that both their corresponding
girth and minimum distance are at least equal to six. The proposed method significantly
reduces the number of sixcycles by shorten the codes and by removing blockcolumns
of the paritycheck matrix in a structured manner.
A QCLDPC code can be counted as one of such algebraic constructions, which
is based on circulant permutation matrices. It is essential to figure out the
proper shift values, which construct no short cycles, since the cycle structures
in QCLDPC codes are ascertained by the shift values of circulant permutation
matrices, either randomly or algebraically (Miladinovic
and Fossorier, 2004; Tanner et al., 2001).
In case of random selection numbers of computations are required to find the
proper shift values, which yield a large girth. As a result, algebraic methods
are desirable to find good shift values. Few promising methods are experienced
to guarantee a large girth and Tanner’s QCLDPC code (Tanner
et al., 2001) is one of such constructions. Tanner
et al. (2001) investigated a class of LDPC block codes, which is
known to guarantee a large girth. These quasicyclic groupstructured regular
LDPC codes have highly symmetric graphs based on simple algebraic description.
When the length of the basic cycle p is large, the graphs appear to have a relatively
large girth for the graph size and vertex degrees, reaching the maximum girth
of twelve for many of the (3, 5) codes. Simulation results show that the proposed
codes perform significantly better than randomly generated (3, 5) LDPC codes
for lengths of 1055 or less. In the random selection of shift values, it acquires
too much computation to figure out the proper shift values, which yield a large
girth. Hence, it is suitable to adopt algebraic methods to locate good shift
values. Kim et al. (2006) examined the cycles
of Tanner (3, 5) QCLDPC codes and derive their girth values. Conditions in
the proposed method are expressed for cycles of different lengths in Tanner
(3, 5) QCLDPC codes as simple polynomial equations in a primitive 15th root
of unity in prime field. Simulation results depict that when p is 31, the girth
of the code is 8, and when p is 61 or 151, the girth of the proposed code is
10. Correspondingly to the (3, 5) case, the other Tanner (J, L) quasicyclic
LDPC codes can also be easily examined. Another class of algebraically structured
QCLDPC codes and their convolutional counterparts is presented by Tanner
and Woodard (2004). Structure of multiplicative groups in the set of integers
modulo m is used to place circulant matrices with a parity check matrix, so
as to shape regular QCLDPC block codes with short to moderate block lengths
and rates and find the performance of proposed codes comparably to random LDPC
codes.
In the work presented by Tanner et al. (2001)
and other researchers, their constructed LDPC codes are from short to moderate
block lengths. Authors replicate the constraint structure of the QCLDPC block
code to infinity and modified construction of the proposed codes to yield irregular
LDPC codes, which do significantly well in the low SNR regime, but some suffer
from poor distance especially, at high SNR due to error floor. They employ search
method to find the girth of graph which acquires too much computation to figure
out the proper shift values and resultantly with many short cycles of length
4. A novel method is required to save the shift value of matrix in order to
save the time and memory. Additionally, a robust method requires finding the
highest girth by avoiding unnecessary short cycles.
Encoding and decoding of QCLDPC based on belief propagation: A novel
methodology for designing structured quasicyclic generalized LDPC (GLDPC)
codes is presented by Liva et al. (2008). A pragmatic
approach for designing good codes is proposed, based on the insertion of powerful
constraint nodes in LDPC bipartite graphs. Approach of the proposed code is
based on the substitution of check nodes in the protograph of a LDPC code with
stronger nodes based, such as, on Hamming codes. It is observed that such a
design approach, extends to lowrate quasicyclic GLDPC codes with outclass
performance in both the error floor and waterfall regions on the AWGN channel.
The decoder uses in the proposed work is the standard beliefpropation algorithm
(MacKay, 1999) with maximum a posteriori decoding at
each variable node and check node. Analysis of the iterative decoding properties
of a GLDPC code’s design is based on Density Evolution (DE) analysis (Luby
et al., 2001; Richardson and Urbanke, 2001b;
Richardson et al., 2001), constituting a controlling
tool for code designers. Another technique for the design of QCLDPC codes is
proposed by Liu and Schniter (2008) based on generalized
combining method. The proposed method designs a much larger class of QCLDPC
codes with similar performance by loosening the condition for ascertaining the
intermediate parameters. In the proposed work a lot of QCLDPC codes with much
less 6cycles and better performance are designed, by permuting the block rows
of the parity check matrices of the component codes. It is shown that the proposed
QCLDPC codes designed by the generalized combining method outperform those
designed by the Chinese Remainder Theorem (CRT) combining method by 0.5 dB at
a BER of 10^{1}. The performance of irregular LDPC codes is investigated
by Ohhashi and Ohtsuki (2004a) with three BP based decoding
algorithms, specifically the Uniformly Most Powerful (UMP) BPbased algorithm,
the normalized BPbased algorithm, and the offset BPbased algorithm on a fast
Rayleigh fading channel by employing density evolution. It is observed from
the study of proposed method that the performance and decoding complexity of
irregular LDPC codes with the offset BPbased algorithm can be very close to
that with the BP algorithm on the fast Rayleigh fading channel. After successful
evolution of irregular LDPC codes, Ohhashi and Ohtsuki (2004b)
then analyze the performance of regular LDPC codes with the normalized BPbased
algorithms on the fast Rayleigh fading channel. Formulas for short and long
regular LDPC codes are derived based on the Probability Density Function (PDF)
of the initial likelihood information and DE for the normalized BPbased algorithm
on the fast Rayleigh fading channel. Performance of the long regular LDPC codes
with the normalized BPbased algorithm in the proposed method outperforms the
BP algorithm and the UMP BPbased algorithm on fast Rayleigh fading channel.
Sullivan et al. (2005) employed seed matrix for
construction of LDPC codes, which is the chain of two relative incidence matrices
for Fano planes and circulant permutation matrices. The effect on decoding performance
is investigated by employing sumproduct algorithm for column weight 3. Authors
show by using BER simulations that large girth codes perform better than those
with lower girths. A new necessary and adequate condition for determining the
girth of QCLDPC codes is derived by Wu et al. (2008)
based on the theory of adjacency matrices, without an explicit enumeration of
cycles. It is shown from simulation results that the obtained codes are often
with performance comparable to the LDPC codes constructed by progressiveedgegrowth
(PEG) algorithm (Hu et al., 2001; Hu
et al., 2002). The performance error for regular QCLDPC and PEGLDPC
codes are plotted with iterative sumproduct decoding (Chung
et al., 2001), with same code rate and observe that the PEGLDPC
codes are frequently with smaller girth than the proposed QCLDPC code by adopting
concentrated paritycheck degree distribution. There is considerable work on
optimizing girth in LDPC codes.
A large girth LDPC codes construction based on linear congruence is proposed
by Jing et al. (2007). Based on graphtheoretic
method, three kinds of special paths are designed in the proposed method to
ensure that the Tanner graph of the parity check matrix mapped from the connection
graph, based on without short cycles. The proposed method is competent of generating
a class of regular QCLDPC codes with a girth of 12 and a minimum Hamming distance
of no less than 24. The simulation results show that the proposed LDPC codes
significantly outperform random codes and the QC block LDPC code with similar
block lengths and rates. Another method for constructing large girth QCLDPC
codes from graphical models is proposed by Huang et al.
(2008). The proposed QCLDPC codes based on circulant permutation matrices
with girths 16 and 18, by employing a simple quadratic congruential equation.
The simulation results show that the proposed codes perform better than the
randomly constructed LDPC codes for short to moderate block lengths and have
almost the same performance as the SridharaFujaTanner (SFT) (Tanner
et al., 2001) codes for different block lengths and rates.
Yang et al. (2008) proposed ParallelInput ParallelOutput
(PIPO) structure for QCLDPC codes to execute a faster and more efficient QCLDPC
encoder than that based on conventional SerialInput ParallelOutput (SIPO)
or ParallelInput SerialOutput (PISO) structure. Experimental results reveal
that the proposed PIPO encoding structure for QCLDPC codes perform significantly
better than the conventional SIPO and PISO structures in terms of speed and
complexity. The proposed scheme shows that the number of LC registers required
by PIPO at the same encoding speed, is linearly proportional to the block size
of the submatrix in the generator matrix. While in case of SIPO and PISO, it
is proportional squarely. Additionally, the LC combinationals required by SIPO
are about 2 to 3 times as much as that PIPO requires. The encoding architecture
is investigated, based on the proposed PIPO structure for multirate QCLDPC
codes, employed by Chinese Digital Television Terrestrial Broadcasting (DTMB),
with less logic complexity compared with SIPO architecture. It is shown from
simulation results that PIPO proposes more flexible tradeoffs between encoding
speed and encoding complexity, which points that the maximal throughput has
the potential over 1Gbps.
With the requirements, defined by MacKay et al.
(2004) for the check matrix of quantum LDPC, Zhou et
al. (2008) proposed a novel construction method of quantum LDPC based
on classical quasicyclic sparse sequence, which can fix both bit flip and phase
shift errors. The results show that quantum LDPC code can correct the bit error
efficiently, and the proposed construction method of quantum LDPC code is available.
It is observed from the proposed work that quantum error correction code based
on classical LDPC code is a promising research area in quantum communications
and quantum computation.
In many ways, LDPC codes can be considered serious competitors to turbo codes.
In particular, LDPC codes show an asymptotically better performance than turbo
codes and they admit a broad range of tradeoffs between performance and decoding
complexity (Richardson and Urbanke, 2001a). The foremost
criticism concerning LDPC codes is their apparent high encoding complexity.
Richardson and Urbanke (2001a) considerd the encoding
problem for codes determined by sparse paritycheck matrices. In the proposed
work the encoding complexity is upper bounded by n+g^{2}, where g ,
the gap, measures in some way to be made specific shortly, the distance
of the given paritycheck matrix to a lower triangular matrix. For the (3, 6)regular
LDPC code, the complexity of encoding is essentially quadratic in the block
length (n). The proposed work shows that even the large block lengths admit
practically feasible encoders, because of the extremely small constant factor.
Hocevar (2003a, b) design encoderdecoder
solution for Irregular Partitioned Permutation (IPP) LDPC codes, for both software
and hardware based on Sridhara, Fuja and Tanner (SFT) method (Tanner
et al., 2001). Two approaches are discussed by Hocevar, first is
the preimplementation phase involving matrix factorization, and the second
approach is the operation and implementations for data encoding. The quasicyclic
factorization or row reduction at the block level in the proposed scheme forms
the core of the preimplementation portion and attains a high degree of parallelism,
thus enabling very high data rate and flexible decoders. It is much faster,
up to two orders of magnitude, and it is simpler than the standard quasicyclic
approach that uses cyclic shifting. The prime advantage of the proposed method
is that it does not need storing a matrix inverse in full binary form that would
require a nontrivial memory, whether exploiting symmetry or not, and the solution
hardware is trivial.
A simple yet low complex systematic LDPC encoding method proposed for class
of QCLDPC codes to let LDPC encoder attains an interchangeable structure, used
in the decoder. With the proposed encoding scheme, implementation of the proposed
encoder becomes much more hardware efficient than having a separate hardware
due to LDPC encoder and decoder resource sharing. Additionally, the overall
computational complexity of the proposed encoding scheme is lower than the wellknown
Richardson’s efficient encoding scheme (Richardson
and Urbanke, 2001a). Authors show that the proposed LDPC encoding scheme
is directly applicable to current the WLAN and WiMAX standards. Another low
complexity and fast encoding schemes, which has also reduced computational complexity
from Richardson's is proposed by Kim et al. (2008).
In that work, authors focus on computational complexity of Richardson's LDPC
matrix, which is composed by matrix A, B, C, D, E and F (Richardson
and Urbanke, 2001b) and propose two schemes for low complexity encoding.
First approaches T = φ and confines D consisting of dual diagonal matrices
and second makes T = φ = 1 without cycle4, resultantly achieve reducing
the complexity to O(n). Proposed encoding schemes are very useful for highrate
and fast communication systems due to reduce complexity and efficiently omitted
processes of encoding.
Authors in the aforesaid work construct QCLDPC codes decoder based on density
evolution in order to compute the threshold of noise level for a large class
of binaryinput channels. But it is not able to estimate their performance in
the case of finite length. Additionally, their interconnection increases the
hardware complexity. The new proposed QCLDPC codes with reduce hardware complexity
because performance of LDPC codes of finite length may be affected by other
elements such as cycle property and minimum distance not only by density evolution.
The codes constructed by the authors in the aforesaid systems are with high hardware complexity which makes them unusable at very large lengths. The unstructured interconnection comes up with routing complexity and obstruction in decoder implementations the number of row connections is almost uniformly distributed by first selecting randomly the rows with the least number of connections. The resultant codes are with rigid row and column weight. The decoders of aforesaid system run with maximum iteration even by achieving the task at early iteration. This will not only waste the time but also degrades the system performance. Moreover, the iterative techniques although improve the decoder throughput but on the other hand memory size is increased.
Abid et al. (2009ad)
keeping in mind the aforesaid system has proposed a novel construction of QCLDPC
code which reduces not only encoding complexity but improve the decoding part
of the system. In order to simplify the hardware implementation, the proposed
codes incorporate some form of structured decoder interconnections. In the proposed
algorithm, the restructuring of the interconnections is invented by splitting
the rows with the group size. Such a division guarantees a concentrated node
degree distribution and reduces the hardware complexity. The new codes offer
more flexibility in terms of high girth, multiple code rates and block length.

REFERENCES 
1: Abid, Y., S. Othman, M.F.M. Salleh and G. Farid, 2009. Lower computation and storage complexity of QCLDPC codes in rayleigh fading channel. Int. J. Comput. Theory Eng., 1: 115118.
2: Abid, Y., S. Othman, M.F.M. Salleh and A. Sardar, 2009. An efficient encodingdecoding of large girth LDPC codes based on quasicyclic. Aust. J. Basic Applied Sci., 3: 17341739.
3: Abid, Y., S. Othman, M.F.M. Salleh and G. Farid, 2009. A new quasicyclic low density parity check codes. Proceedings of IEEE Symposium on Industrial Electronics and Applications, Oct. 46, Kuala Lumpur, Malaysia, pp: 329342
4: Abid, Y., S. Othman, M.F.M. Salleh and G. Farid, 2009. Row division method to generate QCLDPC codes. Proceedings of IEEE 5th Advanced International Conference on Telecommunications, May 2428, Venice, Mestre, Italy, pp:183187
5: Chung, S.Y., T.J. Richardson and R.L. Urbanke, 2001. Analysis of sumproduct decoding of lowdensity paritycheck codes using a gaussian approximation. IEEE Trans. Inform. Theory, 47: 657670. Direct Link 
6: Eleftheriou, E. and S. Olcer, 2002. Lowdensity paritycheck codes for digital subscriber lines. IEEE Int. Conf. Commun., 3: 17521757. Direct Link 
7: Fan, J.L., 2000. Array codes as lowdensity paritycheck codes. Proceedings of the 2nd International Symposium on Turbo Codes and Related Topics, Sept. 47, Brest, France, pp: 545546
8: Fossorier, M.P.C., 2004. Quasicyclic low density parity check codes from circulant permutation matrices. IEEE Trans. Inform. Theory, 50: 17881794. Direct Link 
9: Gallager, R.G., 1963. LowDensity ParityCheck Code. MIT Press, Cambridge, MA., pp: 90 Direct Link 
10: Hocevar, D.E., 2003. LDPC code construction with flexible hardware implementation. Proc. IEEE Int. Conf. Commun., 4: 27082712. Direct Link 
11: Hocevar, D.E., 2003. Efficient encoding for a family of quasicyclic LDPC codes. Proceedings of the IEEE Conference on Global Telecommunication, Dec. 51, Dallas, TX, USA., pp: 39964000 CrossRef  Direct Link 
12: Hu, X.Y. E. Eleftheriou and D.M. Arnold, 2001. Progressive edgegrowth tanner graphs. Proceedings of the IEEE Conference on Global Telecommunications, Nov. 2529, Dallas, pp: 9951001
13: Hu, X.Y. E. Eleftheriou and D.M. Arnold, 2002. Irregular progressive edge growth (PEG) tanner graphs. Proceedings of the IEEE International Symposium on Information Theory, July 3035, Lausanne, Switzerland, pp: 480480
14: Huang, C.M., J.F. Huang and C.C. Yang, 2008. Construction of quasicyclic LDPC codes from quadratic congruences. IEEE Commun. Lett., 12: 313315. Direct Link 
15: Jing, L.J., J.L. Lin and W.L Zhu, 2007. Design of quasicyclic lowdensity parity check codes with large girth. ETRI J., 29: 381389. Direct Link 
16: Kim, J.K., S.P. Balakannan, M.H. Lee and C.J. Kim, 2008. Low complexity encoding of LDPC codes for highrate and highspeed communication. Proceedings of the 1st International Conference on Distributed Framework and Applications, Oct 2122, Penang, pp: 189193 CrossRef  Direct Link 
17: Kim, J.L., U. Peled, I. Perepelitsa and V. Pless, 2002. Explicit construction of families of LDPC codes with girth at least six. Proceedings of the 40th Annual Allerton Conference on Communications, Control and Computing, Oct. 24, Univ. of Illi. at UrbanaChampaign, IL, pp: 10241031
18: Kim, S., J.S. No, H. Chung and D.J. Shin, 2006. On the girth of tanner's (3, 5) quasicyclic LDPC codes. IEEE Trans. Inform. Theory, 52: 17391744.
19: Liu, H. and P. Schniter, 2008. Iterative frequencydomain channel estimation and equalization for singlecarrier transmissions without cyclicprefix. IEEE Trans. Wireless Commun., 7: 36863691. Direct Link 
20: Liva, G., W.E. Ryan and M. Chiani, 2008. Quasicyclic generalized LDPC codes with low error floors. IEEE Trans. Commun., 56: 4957. CrossRef  Direct Link 
21: Luby, M.G., M. Mitzenmacher, M.A. Shokrollahi and D.A. Spielman, 2001. Analysis of low density codes and improved designs using irregular graphs. IEEE Trans. Inform. Theory, 47: 585598. CrossRef  Direct Link 
22: Mackay, D.J.C., 1999. Good errorcorrecting codes based on very sparse matrices. IEEE Trans. Inform. Theory, 45: 399431. CrossRef  Direct Link 
23: MacKay, D.J.C. and M.S. Postol, 2003. Weaknesses of margulis and ramanujanmargulis lowdensity paritycheck codes. Electronic Notes Theoretical Comput. Sci., 74: 97104. CrossRef 
24: MacKay, D., G. Mitchison and P.L. McFadden, 2004. Sparse graph codes for quantum error correction. IEEE Trans. Inform. Theory, 50: 23152330. Direct Link 
25: Miladinovic, N. and M. Fossorier, 2004. Systematic recursive construction of LDPC codes. IEEE Commun. Lett., 8: 302304. Direct Link 
26: Milenkovic, O. and S. Laendner, 2004. Analysis of the cyclestructure of LDPC codes based on Latin squares. Proceedings of the IEEE International Conference on Communications, June 2024, Boulder, CO, USA., pp: 777781 Direct Link 
27: Ohhashi, A. and T. Ohtsuki, 2004. Performance analysis of BPbased algorithms for irregular lowdensity paritycheck codes on fast Rayleigh fading channel. Proceedings of the IEEE 60th Conference on Vehicular Technology, Sept 2629, Chiba, Japan, pp: 25302534 CrossRef  Direct Link 
28: Ohhashi, A. and T. Ohtsuki, 2004. Regular lowdensity paritycheck (LDPC) code with normalized and UMP BPbased algorithms on fast Rayleigh fading channel. Proceedings of the IEEE 60th Conference on Vehicular Technology, Sept 2629, Noda, Japan, pp: 25202524 CrossRef  Direct Link 
29: Richardson, T.J. and R.L. Urbanke, 2001. Efficient encoding of lowdensity paritycheck codes. IEEE Trans. Inform. Theory, 47: 638656. CrossRef  Direct Link 
30: Richardson, T.J. and R.L. Urbanke, 2001. The capacity of low density paritycheck codes under messagepassing decoding. IEEE Trans. Inform. Theory, 47: 599618. CrossRef 
31: Richardson, T.J., A. Shokrollahi and R. Urbanke, 2001. Design of capacityapproaching irregular lowdensity paritycheck codes. IEEE Trans. Inform. Theory, 47: 619637. Direct Link 
32: Sullivan, M.E.O., J. Brevik and R. Wolski, 2005. The performance of LDPC codes with large girth. 43rd Allerton Confertence on Communication Control and Computing. http://pompone.cs.ucsb.edu/~rich/publications/g4.pdf.
33: Tanner, R. and J. Woodard, 2004. WCDMA Requirements and Practical Design. Wiley, Chichester, West Sussex, ISBN: 9780470861776
34: Tanner, R.M., 1981. A recursive approach to low complexity codes. IEEE Trans. Inform. Theory, 27: 533547. Direct Link 
35: Tanner, R.M., D. Sridhara and T.E. Fuja, 2001. A class of groupstructured LDPC codes. Proceedings of the International Conference on Information Systems Technology and its Applications, June 1315, Ambleside, England, pp: 282282
36: Wu, X., X. You and C. Zhao, 2008. A necessary and sufficient condition for determining the girth of quasicyclic LDPC codes. IEEE Trans. Commun., 56: 854857. Direct Link 
37: Yang, Z., Q. Xie, K. Peng and J. Fu, 2008. A fast and efficient encoding structure for QCLDPC codes. Proceedings of the International Conference on Communications, Circuits and Systems, May 1923, Beijing, China, pp: 1620
38: Zhou, B., L. Zhang, J. Kang, Q. Huang, Y.Y. Tai, S. Lin and M. Xu, 2008. Nonbinary LDPC codes vs. ReedSolomon codes. Proceedings of the Workshop on Information Theory and Applications, Jan. 27Feb. 1, San Diego, CA., pp: 175184 CrossRef  Direct Link 
39: Berrrou, C., A. Glavieux and O. Thitimajshima, 1993. Near Shannon limit errorcorrecting coding and decoding: Turbocodes. Proceedings of the IEEE International Conference on Communication, May 2326, 1993, Geneva, Switzerland, pp: 10641070 CrossRef 



