on the minimum distance of generalized spatially coupled...
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On the Minimum Distance of Generalized Spatially Coupled LDPC Codes
Mitchell, David G.M.; Lentmaier, Michael; Costello Jr., Daniel J.
Published in:IEEE International Symposium on Information Theory Proceedings (ISIT), 2013
DOI:10.1109/ISIT.2013.6620551
2013
Link to publication
Citation for published version (APA):Mitchell, D. G. M., Lentmaier, M., & Costello Jr., D. J. (2013). On the Minimum Distance of Generalized SpatiallyCoupled LDPC Codes. In IEEE International Symposium on Information Theory Proceedings (ISIT), 2013 (pp.1874-1878). IEEE - Institute of Electrical and Electronics Engineers Inc..https://doi.org/10.1109/ISIT.2013.6620551
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On the Minimum Distance of GeneralizedSpatially Coupled LDPC Codes
David G. M. Mitchell⇤, Michael Lentmaier†, and Daniel J. Costello, Jr.⇤⇤Dept. of Electrical Engineering, University of Notre Dame, Notre Dame, Indiana, USA,
{david.mitchell, costello.2}@nd.edu†Dept. of Electrical and Information Technology, Lund University, Lund, Sweden
Abstract—Families of generalized spatially-coupled low-densityparity-check (GSC-LDPC) code ensembles can be formed byterminating protograph-based generalized LDPC convolutional(GLDPCC) codes. It has previously been shown that ensemblesof GSC-LDPC codes constructed from a protograph have betteriterative decoding thresholds than their block code counterparts,and that, for large termination lengths, their thresholds coincidewith the maximum a-posteriori (MAP) decoding threshold ofthe underlying generalized LDPC block code ensemble. Herewe show that, in addition to their excellent iterative decodingthresholds, ensembles of GSC-LDPC codes are asymptoticallygood and have large minimum distance growth rates.
I. INTRODUCTION
Low-density parity-check convolutional (LDPCC) codes[1] have been shown to be capable of achieving capacity-approaching performance with iterative message-passing de-coding [2]. The excellent iterative decoding thresholds [3], [4]that these codes display has been attributed to the threshold
saturation effect [5], [6]. In addition to good threshold perfor-mance, it can also be shown that the minimum free distancetypical of most members of these LDPCC code ensemblesgrows linearly with the constraint length as the constraintlength tends to infinity, i.e., they are asymptotically good [7],[8]. A large free distance growth rate indicates that codesrandomly drawn from the ensemble should have a low errorfloor under maximum likelihood (ML) decoding.
Generalized LDPC (GLDPC) block codes were first pro-posed by Tanner [9] and have been shown to possess manydesirable features, such as large minimum distance [10],[11] and good iterative decoding performance [12]. Followingthis construction, more complicated constraints than a singleparity-check (SPC) constraint are permitted. In other words,a constraint node with n inputs can represent an arbitrary(n, k) linear block code. In [13], it was shown that ensem-bles of generalized terminated LDPCC codes, called gen-eralized spatially-coupled LDPC codes (GSC-LDPC) codes,constructed from a protograph have better iterative decodingthresholds than their block code counterparts, and that, forlarge termination lengths, their thresholds coincide with themaximum a-posteriori (MAP) decoding threshold of the un-derlying GLDPC block code ensemble.
In this paper, using weight enumerator evaluation techniquespresented by Abu-Surra, Divsalar, and Ryan [14], we study theasymptotic weight spectrum of GSC-LDPC code ensembles.We show, using a (2, 7)-regular GLDPC block code with(7, 4) Hamming code constraints as an example, that thecorresponding GSC-LDPC code ensembles are asymptoticallygood and have large minimum distance growth rates. As the
termination length increases, we obtain a family of codeswith capacity approaching iterative decoding thresholds anddeclining minimum distance growth rates. However, sincethese are convolutional codes, a more appropriate distancemeasure for assessing the ML decoding performance of suchcode ensembles is the free distance growth rate of the associ-ated ensemble of periodically time-varying generalized LDPCconvolutional (GLDPCC) codes. Consequently, in the finalpart of the paper, we show that the terminated GSC-LDPCcode ensembles can be used to obtain an upper bound onthe free distance growth rate of ensembles of periodicallytime-varying GLDPCC codes. The free distance growth ratecan also be bounded below by using ensembles of tail-bitingGLDPCC codes using a similar technique to one previouslypresented for LDPCC code ensembles with SPC constraints[15], [8]. By comparing and evaluating these bounds we findthat, for a sufficiently large period, the bounds coincide, givingus exact values for the GLDPCC code ensemble free distancegrowth rates.
II. BACKGROUND
A protograph [16] is a small bipartite graph that connectsa set of nv variable nodes V = {v1, . . . , vnv} to a set of nc
generalized constraint nodes C = {c1, . . . , cnc} by a set ofedges E. In a protograph-based GLDPC code, each constraintnode cm can represent an arbitrary block code of length ncm .Figure 1 displays the protograph of a (2, 7)-regular GLDPCblock code.
2vv1 3v 4v 5v 6v 7v
46
7
32
1
55
6 7 12 3
4
Fig. 1: Protograph of a (2, 7)-regular GLDPC block code. The white circlesrepresent generalized constraint nodes and the black circles represent variablenodes. The labels on the edges indicate the corresponding columns of theparity check matrix of the generalized constraint code.
A protograph can be represented by means of an nc ⇥ nv
bi-adjacency matrix B, which is called the base matrix of theprotograph. The entry in row i and column j of B is equal tothe number of edges that connect nodes ci and vj . The basematrix of the protograph in Fig. 1 is given by
B =
1 1 1 1 1 1 11 1 1 1 1 1 1
�. (1)
In order to construct ensembles of protograph-basedGLDPC codes, a protograph can be interpreted as a templatefor the Tanner graph of a derived code, which can be obtainedby a copy-and-permute operation [16]. The protograph is liftedby replicating each node N times and the edges are permutedamong these replicated nodes in such a way that the structureof the original graph is preserved. Allowing the permutationsto vary over all N ! possible choices results in an ensemble ofGLDPC block codes.
A. Convolutional protographs
An unterminated GLDPCC code can be described by aconvolutional protograph [4] with base matrix
B[0,1] =
2
6666664
B0
B1 B0... B1. . .
Bms
.... . .
Bms . . .
3
7777775, (2)
where ms denotes the syndrome former memory of theconvolutional code and the bc ⇥ bv component base matrices
Bi, i = 0, 1, . . . ,ms, represent the edge connections from thebv variable nodes at time t to the bc (generalized) constraintnodes at time t + i. An ensemble of (in general) time-varying GLDPCC codes can then be formed from B[0,1]
using the protograph construction method described above.The decoding constraint length of the resulting ensemble isgiven as ⌫s = (ms + 1)Nbv .
Starting from the base matrix B of a block code ensemble,one can construct GLDPCC code ensembles with the samecomputation trees. This is achieved by an edge spreading
procedure (see [4] for details) that divides the edges from eachvariable node in the base matrix B among ms+1 componentbase matrices Bi, i = 0, 1, . . . ,ms, such that the conditionB0 + B1 + · · · + Bms = B is satisfied. For example, wecould apply the edge spreading technique to the (2, 7)-regularblock base matrix in (1) to obtain the following componentbase matrices
B0 =
0 0 0 0 1 1 11 1 1 0 0 0 0
�, (3)
B1 =
1 1 1 1 0 0 00 0 0 1 1 1 1
�. (4)
From a convolutional protograph with base matrix B[0,1],we can form a periodically time-varying N -fold graph coverwith period T by choosing, for the bc ⇥ bv submatricesB0,B1, . . . ,Bms in the first T columns of B[0,1], a set ofN ⇥N permutation matrices randomly and independently toform Nbc⇥Nbv submatrices H0(t),H1(t+1), . . . ,Hms(t+ms), respectively, for t = 0, 1, . . . , T � 1. These submatricesare then repeated periodically (indefinitely) to form a convo-lutional parity-check matrix H[0,1] such that Hi(t + T ) =Hi(t), 8i, t. An ensemble of periodically time-varying GLD-PCC codes with period T , rate R = 1�NMCbc/Nbv = 1�MCbc/bv , and decoding constraint length ⌫s = N(ms + 1)bvcan then be derived by letting the permutation matrices used toform H0(t),H1(t+1), . . . ,Hms(t+ms), for t = 0, 1, . . . , T�1, vary over the N ! choices of an N ⇥N permutation matrix.
III. TERMINATION OF GLDPCC CODES
Suppose that we start the convolutional code with parity-check matrix defined in (2) at time t = 0 and terminate itafter L time instants. The resulting finite-length base matrixis then given by
B[0,L�1] =
2
6666664
B0...
. . .Bms B0
. . ....
Bms
3
7777775
(L+ms)bc⇥Lbv
. (5)
The matrix B[0,L�1] can be considered as the base matrix ofa terminated protograph-based GLDPCC code, or generalizedspatially-coupled LDPC (GSC-LDPC) code. This terminatedprotograph is slightly irregular with lower constraint nodedegrees at the beginning and end. These shortened constraintnodes are now associated with shortened constraint codes inwhich the symbols of the missing edges are removed. Note thatsuch a code shortening is equivalent to fixing these removedsymbols and assigning an infinite reliability to them. Thevariable node degrees are not affected by termination.
The parity-check matrix H of the block code derived fromB[0,L�1] by lifting with some factor N has NbvL columnsand (L+ms)NbcMC rows, where MC denotes the number ofparity-checks of the constraint code.1 It follows that the rateof the GSC-LDPC code is equal to
RL = 1� (L+ms)bcMC ��
Lbv(6)
for some � � 0 that accounts for a slight rate increase dueto the shortened constraint nodes. If H has full rank, the rateincrease parameter is given by � = 0. However, shortenedconstraint codes at the ends of the graph can cause a reducedrank for H, which slightly increases RL. In this case, � > 0and depends on both the particular constraint code chosen andthe degree of shortening. As L ! 1, the rate RL convergesto the rate of the underlying GLDPC block code with basematrix B.
The generalized convolutional base matrix B[0,1] can alsobe terminated using tail-biting [17], [18], resulting in a gen-eralized tail-biting LDPC (GTB-LDPC) code. Here, for any� � ms, the last bcms rows of the terminated parity-checkmatrix B[0,��1] are removed and added to the first bcms rowsto form the �bc⇥�bv tail-biting parity-check matrix B
(�)tb with
tail-biting termination factor �:2
666666666666666664
B0 Bms · · · B1
B1 B0. . .
......
... Bms
Bms Bms�1
Bms
. . .B0
. . .... B0
. . . Bms�1
.... . .
Bms Bms�1 · · · B0
3
777777777777777775
.
(7)1We assume here that each generalized constraint node in the block
protograph is of the same type and has MC parity-checks. This assumptioncan be relaxed in general.
123 4
567
567
1234
123412
345
67
567
567
567
567
567
123 4
1234
123 4
1234
t=0 t=2 t=L-2 t=L-1 t=L
Fig. 2: Protograph of a (2, 7)-regular GLDPCC code ensemble. The white circles represent generalized constraint nodes and the black circles represent variablenodes.
Note that, if ms = 1 and � = 1, the tail-biting base matrixis simply the original block code base matrix, i.e., B(1)
tb = B.Terminating B[0,1] in such a way preserves the design rate ofthe ensemble, i.e., R� = 1��bcMC/�bv = 1�bcMC/bv = R,and we see that B(�)
tb has exactly the same degree distributionas the original block code base matrix B.
IV. MINIMUM DISTANCE ANALYSIS OFPROTOGRAPH-BASED GSC-LDPC CODE ENSEMBLES
In [14], Abu-Surra, Divsalar, and Ryan presented a tech-nique to calculate the average weight enumerator and asymp-totic spectral shape function for protograph-based GLDPCcode ensembles. The spectral shape function can be used totest if an ensemble is asymptotically good, i.e., if the minimumdistance typical of most members of the ensemble is at least aslarge as �minn, where �min is the minimum distance growth
rate of the ensemble and n is the block length.Consider the protograph with generalized constraint nodes
shown in Fig. 1. If we suppose the constraint nodes to be (7, 4)Hamming codes with parity-check matrix
H1 =
2
41 0 0 1 1 1 00 1 0 1 1 0 10 0 1 1 0 1 1
3
5 ,
then the resulting ensemble has design rate R = 1/7, isasymptotically good, and has growth rate �min = 0.186 [14].
We will construct the base matrix of a GSC-LDPC codeensemble using (2) and component base matrices (3) and (4).The resulting protograph is shown in Fig. 2. We use the(7, 4) Hamming code with parity-check matrix H1 for thegeneralized constraint nodes. The numbers on the edges ofthe protograph in Fig. 2 indicate which columns of H1 (orshortened version of H1) the nodes are connected to. Aftertermination, the resulting ensemble corresponds to a GSC-LDPC code ensemble. The design rate of the ensemble is givenas
RL = 1� 6(L+ 1)� 2
7L. (8)
Note that � = 2 in this example because the two leftmost(shortened) constraint nodes in Fig. 2 correspond to shortenedcodes with rate 1/3, i.e., the number of parity-checks in thesetwo constraint nodes is MC = 2, while all of the otherconstraint nodes have MC = 3 parity-checks. These ensembleswere shown to have thresholds numerically indistinguishablefrom the MAP threshold as L ! 1 in [13].
The evaluation of the asymptotic weight enumerators forGSC-LDPC codes is complex, since the conjecture regardingsimplification of the numerical evaluation proposed in [14]
cannot immediately be applied to these ensembles. This con-jecture relies on grouping together nodes of the same typeand optimizing them together. However, in the GSC-LDPCcase, nodes from different time instants must be optimizedseparately, even if they are of the same type.
Fig. 3 shows the asymptotic spectral shape functions forthe GSC-LDPC code ensembles with termination factors L =7, 8, 10, 12, 14, 16, 18, and 20. Also shown are the asymptoticspectral shape functions for “random” codes of correspondingrate RL calculated using (see [19])
r(�) = H(�)� (1�RL) ln(2), (9)
where H(�) = �(1� �) ln(1� �)� � ln(�). We observe thatthe GSC-LDPC code ensembles are asymptotically good andhave large minimum distance growth rates. This indicates thata long code based on this protograph has, with probability nearone, a large minimum distance. As L increases, the design rateincreases and the minimum distance growth rate decreases.This behavior is the same as was observed in the SPC case[20].
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5!0.04
!0.02
0
0.02
0.04
0.06
0.08
0.1
!
r(!)
Random codesof rate R
L
GSC!LDPCL = [20,18,16,14,12,10,8,7]
Fig. 3: Spectral shape functions of GSC-LDPC code ensembles and randomlinear codes of the corresponding rate.
V. FREE DISTANCE ANALYSIS OF PROTOGRAPH-BASEDGLDPCC CODE ENSEMBLES
In Fig. 3 we saw that the minimum distance growth ratesof GSC-LDPC codes decrease as the termination factor Lincreases. However, since GSC-LDPC codes are terminatedGLDPCC codes, a more appropriate distance measure forassessing the ML decoding performance of such codes is thefree distance growth rate of the GLDPCC ensemble. In thissection, we first calculate the minimum distance growth rates
for GTB-LDPC code ensembles and show that for sufficientlylarge termination factors, the growth rates coincide with thosecalculated for the GSC-LDPC code ensembles in Section IV.We then show that the growth rates of the GTB-LDPC codeensembles and GSC-LDPC code ensembles can be used toobtain lower and upper bounds on the free distance growthrate of the GLDPCC code ensemble, respectively.
A. Minimum distance analysis of GTB-LDPC code ensembles
We now consider terminating the protograph in Fig. 2 asa GTB-LDPC code with termination factor �. Unlike theprevious termination technique, this results in a (2, 7)-regularprotograph with design rate R� = 1/7 for all �. The minimumdistance growth rates of the GTB-LDPC code ensembles arepresented in Fig. 4 alongside those corresponding to the GSC-LDPC code ensembles. We observe that the growth ratesremain constant at �min = 0.186 (the growth rate of theoriginal GLDPC block code ensemble) for � = 1, 2, . . . , 8,and then begin to decay to zero as � ! 1. Also, as a resultof the convolutional structure, we observe that the GTB-LDPCand GSC-LDPC growth rates coincide for L,� � 10. This isthe same behavior that we observed for TB-LDPC and SC-LDPC codes with SPC constraints [8].
B. Free distance bounds for GLDPCC code ensembles
Consider an ensemble of periodically time-varying GLD-PCC codes with rate R = 1 � bcMC/bv and period Tconstructed from a convolutional protograph with base ma-trix B[0,1] (see (2)) as described in Section II-A. Using amodification of the proof techniques in [8], [15], it is possibleto show that the average free distance of this ensemble canbe bounded below by the average minimum distance of anensemble of GTB-LDPC codes derived from the base matrixB
(�)tb (see (7)) with termination factor � = T . Here, we show
that the average free distance of the GLDPCC ensemble canalso be bounded above by the average minimum distance of theensemble of GSC-LDPC codes derived from the base matrixB[0,L�1] (see (5)) with termination factor L = T .
Theorem 1: Consider a rate R = 1 � bcMC/bv untermi-nated, periodically time-varying GLDPCC code ensemble withmemory ms, decoding constraint length ⌫s = N(ms + 1)bv ,and period T derived from B[0,1]. Let d
(L)min be the average
minimum distance of the GSC-LDPC code ensemble withblock length n = LNbv and termination factor L. Thenthe ensemble average free distance d
(T )free of the unterminated
convolutional code ensemble is bounded above by d(L)min for
termination factor L = T , i.e.,
d(T )free d
(T )min. (10)
Sketch of proof. There is a one-to-one relationship be-tween members of the periodically time-varying GLD-PCC code ensemble and members of the correspondingGSC-LDPC code ensemble with termination factor L =T . For any such pair of codes, every codeword x =[ x0 x1 · · · xLNbv�1 ] in the GSC-LDPC (terminatedconvolutional) code can also be viewed as a codewordx[0,1] = [ x0 x1 · · · xLNbv�1 0 · · · ] in the unter-minated code. It follows that the free distance d
(T )free of the
unterminated code cannot be larger than the minimum distance
d(T )min of the terminated code. The ensemble average result
d(T )free d
(T )min then follows directly. 2
Since there is no danger of ambiguity, we will henceforthdrop the overline notation when discussing ensemble averagedistance measures.
C. Free distance growth rates of GLDPCC code ensembles
For GLDPCC codes, it is natural to define the free distancegrowth rate with respect to the decoding constraint length ⌫s,i.e., as the ratio of the free distance dfree to ⌫s.
By bounding d(T )free using (10), we obtain an upper bound
on the free distance growth rate as
�(T )free =
d(T )free
⌫s �
(T )minT
(ms + 1), (11)
where �(T )min = d
(T )min/n = d
(T )min/(NTbv) is the minimum
distance growth rate of GSC-LDPC code ensemble with ter-mination factor L = T and base matrix B[0,T�1]. Similarly,using a similar argument to that presented in [8], we have
�(T )free �
�(T )minT
(ms + 1), (12)
where �(T )min is the minimum distance growth rate of the GTB-
LDPC code ensemble with tail-biting termination factor � = T
and base matrix B(�)tb .
The free distance growth rate �(T )free that we bound from
above using (11) is, by definition, an existence-type lowerbound on the free distance of most members of the ensemble,i.e., with high probability a randomly chosen code from theensemble has minimum free distance at least as large as�(T )free⌫s as ⌫s ! 1. Note that the free distance growth rate
may also be calculated with respect to the encoding constraintlength ⌫e, which corresponds to the maximum number oftransmitted symbols that can be affected by a single nonzeroblock of information digits. As a result of normalizing by thedecoding constraint length, it is possible to have free distancegrowth rates larger than 0.5. For further details, see [8].
D. Numerical results
As an example, we consider once more the (2, 7)-regularGLDPCC code ensemble with memory ms = 1 and rate R =1/7 depicted in Fig. 2. For this case, we calculate the upperbound on the free distance growth rate of the periodically time-varying GLDPCC code ensemble as �
(T )free �
(T )minT/2 using
(11) for termination factors L = T � 7. Fig. 4 displays theminimum distance growth rates �
(L)min of the GSC-LDPC code
ensembles defined by B[0,L�1] for L = 7, 8, 10, 12, . . . , 20that were calculated using the technique proposed in [14] andthe associated upper bounds on the GLDPCC code ensemblegrowth rates �(T )
free �(T )minT/2 for L = T . Also shown are the
minimum distance growth rates �(�)min of the GTB-LDPC code
ensembles defined by base matrix B(�)tb for � = 1, 2, 4, . . . , 20
and the associated lower bounds on the GLDPCC code en-semble growth rates �
(T )free � �
(T )minT/2 for � = T calculated
using (12).We observe that the calculated GTB-LDPC code ensemble
minimum distance growth rates �(�)min remain constant for
� = 1, . . . , 8 and then start to decrease as the termination
0 2 4 6 8 10 12 14 16 18 200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
L or !
Gro
wth
ra
te
GTB!LDPC growth rates "
min(!)
Lower bound on "
free
Upper bound on "
free
GSC!LDPC growth rates "
min(L)
Fig. 4: Minimum distance growth rates of GSC-LDPC code ensembles andGTB-LDPC code ensembles and calculated upper and lower bounds onthe free distance growth rates of the associated periodically time-varyingGLDPCC code ensembles.
factor � grows, tending to zero as � tends to infinity. Cor-respondingly, as � exceeds 8, the lower bound calculated for�(T )free levels off at �(T )
free � 0.805. The calculated GSC-LDPCcode ensemble minimum distance growth rates �(L)
min are largerfor small values of L (where the rate loss is larger) anddecrease monotonically to zero as L ! 1. Using (11) toobtain an upper bound on �
(T )free we observe that, for T � 10,
the upper and lower bounds coincide, indicating that, for thesevalues of the period T , �(T )
free = 0.805, significantly larger thanthe underlying GLDPC block code minimum distance growthrate �min = 0.186. In addition, we note that, at the point wherethe upper and lower bounds on �
(T )free coincide, the minimum
distance growth rates for both termination methods also coin-cide. Recall that the GTB-LDPC code ensembles all have rate1/7, wheras the rate of the GSC-LDPC code ensembles is afunction of the termination factor L given by (8). This generaltechnique can be used to bound the free distance growth rateabove and below for any regular or irregular periodically time-varying protograph-based GLDPCC code ensemble.
While large free distance growth rates are indicative ofgood ML decoding performance, when predicting the iterativedecoding performance of a code ensemble in the high SNRregion other graphical objects such as trapping sets, pseu-docodewords, absorbing sets, etc., come into effect. Based onresults from the SPC case [8], we would expect GSC-LDPCcodes with large minimum/free distance growth rates to alsohave large trapping set growth rates, indicating good iterativedecoding performance in the high SNR region.
VI. CONCLUSIONS
GSC-LDPC codes constructed from a protograph are knownto have better iterative decoding thresholds than their blockcode counterparts, and, for large termination lengths, theirthresholds coincide with the MAP decoding threshold of theunderlying GLDPC block code ensemble. In this paper, weused an asymptotic weight enumerator analysis to show thatGSC-LDPC code ensembles are also asymptotically good.We saw, using a (2, 7)-regular GLDPC code as an example,that the corresponding GSC-LDPC code ensembles have largeminimum distance growth rates for all computed values of L.
This indicates that long codes chosen from these ensembleshave, with probability near one, large minimum distancesas well as excellent iterative decoding thresholds. Finally,we obtained asymptotic minimum distance growth rates forGTB-LDPC code ensembles and showed that the growthrates of GTB-LDPC and GSC-LDPC code ensembles can beused to obtain lower and upper bounds, respectively, on thefree distance growth rate of the associated periodically time-varying GLDPCC code ensemble.
ACKNOWLEDGMENT
This work was partially supported by NSF Grant CCF-1161754.
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