Analysis of Non-binary Hybrid LDPC Codes

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by  Lucile Sassatelli

Analysis of Non-binary Hybrid LDPC Codes
Lucile Sassatelli and David Declercq
ETIS
ENSEA/UCP/CNRS UMR-8051
95014 Cergy, FRANCE
{sassatelli,declercq}@ensea.fr
Abstract—This paper is eligible for the student paper award. II. THECLASS OF HYBRID CODES
In this paper, we analyse asymptotically a new class of LDPC
We define a Non-binaryHybridLDPC code as LDPC code
8 codes called Non-binary Hybrid LDPC codes, which has been
0 recently introduced in [7]. We use density evolution techniques whose variable nodes belong to finite sets of different orders.
0 to derive a stability condition for hybrid LDPC codes, and More precisely, this class of codes is not defined in a finite
2 prove their threshold behavior. We study this stability condition field,butinfinitegroups.Wewillonlyconsidergroupswhose
n to conclude on asymptotic advantages of hybrid LDPC codes cardinality qk is a power of 2, that says groups of the type
a compared to their non-hybrid counterparts. G(q ) = Z pk with p = log (q ). Thus to each element
J of Gk(q ) co2rZrespondsa bkinary m2apkof p bits. Let us call the
5 I. INTRODUCTION minimukm(cid:0)orde(cid:1)rofcodewordsymbolsq k ,andthemaximum
min
1
Like Turbo Codes, LDPC codes are pseudo-random codes order of codeword symbols q . The class of hybrid LDPC
which are well-known to be channel capacity-approaching. codes is defined on the promdauxct group Z pmin × ... ×
T] LDPC codes have been rediscovered by MacKay under their Z pmax.LetusnoticethatthistypeofLDP2ZCcodesbuilton
2Z (cid:0) (cid:1)
I binary form and soon after their non-binary counterpart have productgroupshasalreadybeenproposedintheliterature[2],
s. been studied by Davey [1]. Non-binary LDPC codes have b(cid:0)ut(cid:1)no optimization of the code structure has been proposed
c
[ recently received a great attention because they have better and its application was restricted to the mapping of the
performance than binary LDPC codes for short block length codewordsymbolstodifferentmodulationorders.Paritycheck
1
and/or high order modulations [3], [8], [4]. However, good codes defined on (G(q )×...×G(q )) are particular
v min max
shortlengthnon-binaryLDPC codestend to be ’ultra-sparse’, since they are linear in Z , but could be non-linear in the
3 2Z
3 and have worse convergence threshold than binary LDPC product group. Although it is a loss of generality, we have
2 codes. decided to restrict ourselves to hybrid LDPC codes that are
2 Our main motivation in introducing and studying the new linear in their product group, in order to bypass the encoding
.
1 class of hybrid LDPC codes is to combine the advantages of problem. We will therefore only consider upper-triangular
0 both families of codes, binary and non-binary. Hybrid codes parity check matrices and a specific sort of the symbolorders
8
families aim at achieving this trade-off by mixing different in the codeword, which ensures the linearity of the hybrid
0
order for the symbols in the same codeword. Our resulting codes.Thestructureofthecodewordandtheassociatedparity
:
v codes are called Non-binary Hybrid LDPC Codes because of checkmatrixisdepictedinFigure1.Wehierarchicallysortthe
i
X the mixture of different symbol sets in the codeword.
Redundancy Information
In [7], we have demonstrated the interest of the Hybrid
r
a LDPC codes by designing codes that compare favorably with c= G(qmax) ... G(qr+1) G(qr) ... G(qmin)
existingcodesforquitemoderatecodelength(afewthousands
bits). Hybrid LDPC codes appear to be especially interesting
for low rate codes, R ≤ 0.25. In this paper, we study the G(qmax)
asymptotic behavior and properties of Hybrid LDPC codes
under iterative belief propagation (BP) decoding.
.
.
Thesectiontwoofthispaperhighlightsthegeneralityofour H= .
new codes structure, and explains why we have focused the
asymptotic study on the particular subclass of linear codes. 0
In third and fourth sections, we present the context of the G(qr+1)
study, and detail symmetry and linear-invariance properties
which are useful for the stability condition.This condition is
then expressed and analyzed to show theoretic advantages of Fig.1. Hybridcodeword andparity-check matrix.
Hybrid LDPC codes.
different group orders in the rows of the parity-check matrix,
and also in the codeword, such that q < ... < q <
1This workwassupported bytheFrench ArmamentProcurement Agency min k
(DGA). ... < qmax. To encode a redundancy symbol, we consider
eachsymbolthatparticipatesintheparitycheckasanelement is injective since dim(Im(A))=rank(A)=p ).
k
of the highest group, which is only possible if the groups are
A: G(2)pk → G(2)pl
sorted as in Figure 1. This clearly shows that encoding is
feasible in linear time by backward computation of the check i → j
symbols.
In order to explain the decoding algorithm for hybrid LDPC i denotes the binary map of i in G(qk) in G(2)pk, with
p = log (q ). That is, each index i is taken to mean the
codes, it is useful to interpreta parity check of a hybridcode k 2 k
ith element of G(q ), given some enumeration of the field.
as a special case of a parity check built on the highest order k
x is the ith element of vector x. The extension y, of the
group of the symbols of the row, denoted G(q ) and have a i
l probabilityvectorxby A, isdenotedby x×A anddefinedby:
lookatthe binaryimage ofthe equivalentcode[8]. Forcodes
for all i=0,...,q −1
definedoverGaloisfields,thenonzerovaluesofHcorrespond l
to the companionmatrices of the finite field elements and are if i∈/ Im(A), y =0
i
typically rotation matrices (because of the cyclic property of
if i∈Im(A), y =x with j such that i=Aj
i j
the Galois fields).
In the case of hybrid LDPC codes, a nonzero value is a A is called extension,and the inversefunctionA−1 trunca-
functionthatconnectsa rowin G(ql) and a columnin G(qk), tion from Im(A) to G(qk). The truncation is defined by
i.e., that maps the q symbols of G(q ) into a subset of
k k
q symbols that belongs to G(q ). Such application is not
k l A−1 : Im(A) → G(2)pk
necessarily linear, but in the case it is, its equivalent binary
representation is a matrix of dimension (p ×p ). Note that, j → i with j such that j=Ai
l k
with the above mentioned constraints, we have necessarily
ThetruncationxoftheprobabilityvectorybyA−1isdenoted
pk ≤ pl. It is possible to generalize the Belief propagation by y×A−1 and defined by
decoder to hybrid codes, and it has been shown that even
for those very specific structures, it is possible to derive a i=0,...,q −1, x =y with j such that j=Ai
k i j
fast version of the decoder using FFTs [5]. In this work, we
Given a probability-vector x of size q, the components of
consideronlymapsthatarelinearapplications,andhencethat
the logarithmic density ratio (LDR) vector w associated with
have a binary representation, in order to be able to apply all
known results on linear codes. We call the message passing x are defined as wi =log xx0i , i=0,...,q−1.
stepthroughhij (cf.figure2)extensionwhenitisfromG(qk) At channel output, LDR (cid:16)mes(cid:17)sages are actually logarithmic
to G(q ) and truncation when it is from G(q ) to G(q ). likelihood ratio (LLR) vectors.
l l k
B. Parameterization of Hybrid LDPC family
c1 ∈G(q1) c2 ∈G(q2) c3 ∈G(q3)
An edge of the Tanner graph of an Hybrid LDPC code
2 4 8 has fourparameters(i,qk,j,ql). A hybridLDPC code is then
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) represented by π(i,j,k,l) which is the proportion of edges
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) q1≤q2≤q3 connecting variable nodes of degree i in G(q ), to check
k
hi1(c1) hi2(c2) hi3(c3) nodes of degree j in G(ql). Thus, hybrid LDPC codes have a
very rich parameterizationsince the parameter space has four
dimensions.
8
8 8
C. Symmetry definition for density evolution approach
LetWbeaLLRvectorcomputedattheoutputofadiscrete
parity-checkinG(q3)
memoryless channel, and v the component of the codeword
sent, corresponding to the required value for the data node
hi1(c1)+hi2(c2)+hi3(c3)=0, hij(cj)∈G(q3) the edge with message W is connected to. Wa denotes the
definesacomponentcodeinthegroupG=G(q1)×G(q2)×G(q3) cyclic-permutation of W. c denotes the value of the symbol
linked to the edge with the message W, and y the available
Fig.2. Parity-check ofanhybridLDPCcode. information on all other edges of the graph. Wa is the ith
i
III. PROPERTIES OF LINEARHYBRID LDPCCODES componentof Wa and is definedby Wia =log PP((aa··cc==0i||yy)) ,
where · denotes the multiplication in G(q). Like(cid:16)in [4], W(cid:17)a,
A. The Extension and Truncation Operations
for all a∈G(q) is defined by
We first clarify the nature of the non-zero elements of the
parity-check matrix of a hybrid LDPC code. We consider an Wia =Wa+i−Wa, ∀i=0...q−1
elementAofthesetoflinearextensionsfromG(q )toG(q ).
k l A channel is cyclic if output LLR vector W fulfills
Im(A) denotes the image of A. A belongs to the set of linear
applications from G(2)pk to G(2)pl which are full-rank (that P(Wa|v =0)=P(W|v =a)
Definition 1 On a cyclic channel, a LDR message is symmet- variable, we define the random-extension of size q of X,
2
ric, if the following expression holds denoted X˜, as the probability-vector random variable X×A,
where A is uniformly chosen in E and independent on X.
∀a∈G(q), P(W=w|v =a)=e−waP(W=w|v =0)
Lemma 4: A probability-vector random variable Y is LA-
Most practical channels are cyclic, and thus, in this work, invariantifandonlyifthereexistsaprobability-vectorrandom
we assume transmission on arbitrary memoryless cyclic- variable X such that Y =X˜.
symmetric channels. The generalization of the results in this For lack of space reason, we will detail the proof of this
paper to non-symmetric channels can be done thanks to lemma,whichiseasy,inafuturepublication.Thankstolemma
the coset approach as in [4]. The symmetry property is the 4, the check node incoming messages are LA-invariantin the
essential condition for any asymptotic study since it ensures codefamilymadeofallthepossiblecycle-freeinterleaversand
thatthe errorprobabilityis independentof the codewordsent. uniformly chosen linear extensions (and hence corresponding
Lemma 1: If W is a symmetric LDR-vectorrandomvariable, truncations). Moreover, random-truncations, at check node
then its extension W×A, by any linear extension A with full output,ensuresLA-invarianceofvariablenodeincomingmes-
rank, is also symmetric. sages. Thus, as shown in [7] under Gaussian approximation,
The same lemma holds for truncation. The data pass and the thedensitiesofvectormessagesarecharacterizedbyonlyone
check pass of belief propagation have already been shown parameter.
to preserve symmetry. Thus, lemma 1 ensures that the hybrid
IV. THE STABILITYCONDITION FORHYBRID LDPC
decoderpreservesthesymmetrypropertyiftheinputmessages
CODES
are symmetric.
The stability condition, introduced in [6], is a necessary
Lemma 2: The error-probabilityof a code in a hybrid family,
and sufficient condition for the error probability to approach
used on a cyclic-symmetric channel, is independent on the
arbitrarilyclosetozero,assumingithasalreadydroppedbelow
codeword sent.
somevalueatsome iteration.In thisparagraph,we generalize
For lack of space, we do not give the proof of this lemma,
the stability condition to hybrid LDPC codes.
which is a direct generalization of [6].
Given a hybrid family defined by π(i,j,k,l), we define the
D. Linear Application-Invariance following family parameter:
Now we introduce a property that is specific to the hybrid q −1
k
codes families. Bennatan et al. in [4] used permutation- Ω = π(i=2,k,j,l) (j−1)
q −1
l
invariancetoderiveastabilityconditionfornon-binaryLDPC jX,k,l
codes, and to approximate the densities of graph messages Also for a given memoryless symmetric output channel with
using one-dimensional functionals, for extrinsic information transition probabilities p(y|x) and a mapping δ(·), we define
transfert (EXIT) charts analysis. The difference between non- the following channel parameter:
binary and Hybrid LDPC codes holds in the non-zeros ele-
ments of the parity-check matrix. Indeed, they do not corre- 1 qk−1
∆= π(k,l) p(y|δ(i))p(y|δ(0))dy
spondanymoretocyclicpermutations,buttolinearextensions q −1
l
k,l i=1 Z
ortruncations,thatwedenotebylinearapplications.Thegoal X X p
E.g., for BI-AWGN channel, we have
istoprovethatlinearapplication-invariance(shortenedbyLA-
invariance)of messagesis inducedby choosinguniformlythe 1 qk−1 1
linearextensionswhicharethenon-zeroelementsofthehybrid ∆= π(l,k) exp(− n )
q −1 2σ2 i
parity-check matrix. In particular, LA-invariance allows to l,k l i=1
X X
characterizemessagedensitieswith onlyonescalar parameter
wheren isthenumberofonesinthebinarymapofi∈G(q ).
i k
[7].
Theorem: Let assume (π,δ) given for a hybrid LDPC set.
We work with probabilityvectorrandomvariables,but all the
Let P denotes the probability distribution function of initial
definitions and proofs given in the remaining also apply to 0
messages R(k)(0) for all k. Let Pt = P (R ) denotes the
LDR-vector random variables. We denote by E the set of e e t
averageerrorprobabilityatiterationtunderdensityevolution.
linear extensions from G(q ) to G(q ), and by T the set
of ”inverse functions” of E1, what we2call the set of linear • If Ω ≥ ∆1, then there exists a positive constant ξ =
ξ(π,P ) such that Pt >ξ for all iterations t.
truncations from G(q ) to G(q ) (see previous section on 0 e
linear extensions). 2 1 • If Ω < ∆1, then there exists a positive constant ξ =
ξ(π,P ) such that if Pt < ξ at some iteration t, then
Definition 2: Y is LA-invariant if and only if for all 0 e
(A−1,B−1)∈T×T,the probability-vectorrandomvariables Pet approaches zero as t approaches infinity.
Y×A−1 and Y×B−1 are identically distributed. Forlackofspacereason,wegivethereonlya sketchofthe
Lemma 3: If a probability-vector random variable Y of size proof.
q is LA-invariant, then for all (i,j) ∈ G(q )×G(q ), the Proof
2 2 2
random variables Yi and Yj are identically distributed. • We first give the general lines of the proof of the
Definition 3: Let X be a q -sized probability-vector random necessarycondition.Let R(k) denotesthe variable node
1 t+n
outcoming messages in G(q ) at iteration t+n, where Hence, we find E −1R′ = ∆. This last equation
k 2 1
n = 0,1,.... Since we consider only cyclic-symmetric combined with equation (1) leads to the conclusion that
(cid:0) (cid:1)
channels,wecanapplylemma4from[4]. Itensuresthat P (Q(k)) is lower boudedby a strictly positive constant,
e n
there exists an erasurized channel such that the cyclic- asn tendsto infinity,as soonasΩ∆≥1.Thiscondition
symmetricchannelis a degradedversionof it, and hence is the same for all k. Thus, the necessary condition for
provides a lower bound on the error probability. Let stability is Ω< 1.
Rˆt(+k)n,n=0,1,...,denotetherespectivemessagesofthe • We give now the∆main steps for the proof of the suffi-
erasurized channel, and ǫˆ0 the erasure probability. In the ciencyofthecondition.X(k) denotesaprobability-vector
remainder of the proof, we switch to log-density repre- random variable of size k. We define D and D :
n a
sentationofmessages.LetRˆ′(k) denotetheLDR-vector
t+n
trheeprdeissetnritbautitoionnooffRRˆˆ(t′+k(k)n),.nP=(k)0d,e1n,o.t.e.s.tQhe(nkd)i(swtri)budteinoontoesf Dn(X(k)) = EvuXXi((kk))=EvuXX1((kk))
t+n 0 u 0 u 0
theinitialmessageR′(k)ofthecyclic-symmetricchannel. t  t 
The overline notation0X applied to vector X represents = π(l|k) 1 qk−1E Xi(k)
the vector resulting from random extension followed by l ql−1 i=1 vuX0(k)
randomtruncationof X. Providedthatrandomextension X X u
t 
and truncation are such that X and X are of same size, 1 ql−1 X(l)
wifeQcn(akn) sishotwhetdhiasttrtihbeuteirornoropfrRoˆb′(ta+kb)nil,itiweseahraeveequal. Thus, Da(X(l)) = ql−1 Xj=1 EvuutXj0(l)
 
Toshortenthenotationswecanomittheindexofiteration
Pe(Qn)= π(k)Pe(Q(nk))= π(k)Pe(Q(nk)) t. The data pass is translated by
k k
X X i−1
ThereforePe(Qn)islowerboundedbyaconstantstrictly Ri(k) =Ri(0)(k) L(ik)
greater than zero if and only if there exists k such that n=1
Y
Pe(Qn(k)) is lower bounded by a constant strictly greater We obtain
than zero. Defining
i−1
Ωk = π(i=2,j,l|k)qqkl−−11(j−1) Dn(Rt) = Ei,k0B@vuuutRR((i0kk))1CA=∆Xk π(k)24E0@vuutLL(i(0kk))1A35
j≥2,l
X
First, we are going to prove the recursive inequality (2)
and P = π(k)P(k), we show that We show the three following equations.
0 k 0
P (Q(k))≥P 1 ǫˆ(k)2Ω Ωn−1P n−1 (1) L(k)
e n 2(qmax−1)2 0 k 0 EvuLi(0k)=Dn(L(k))
We prove that P is symmetric in the binary sense, and u
0 t 
as in [4], we obtain D (L(k))= π(l|k)qk−1D (L(l))
n a
q −1
1 1 l l
lim logP (W ≤0)⊗n =log E − R′ X
n→∞n 0 1 2 1
(cid:18) (cid:18) (cid:19)(cid:19) 1−D (L(l))≥ π(j|l)(1−D (R(l)))j−1
whereR′ istheshortenednotationforthefirstcomponent a j a
1 +O(D (R(l))2)
of the mixture of decoder input LLR-vector random P a
variables R′0(k). We have ConnectingDa(R(l)) to Dn(Rt) ends up with the proof
of equation 2:
1 R×A×B−1
E − R′ =E E 1 |A,B
(cid:18) 2 1(cid:19) A,B sR0×A×B−1 !! D (R(l))= 1 π(k|l)qk−1E Ri(k)
and finally obtain a ql−1 k i=1 vuR0(k)
X X u
t 
E(cid:18)−12R1′(cid:19)=Xk,l π(k,l)ql−1 1qXik=−11E rRR0i! Dn(Rt)= l π(l)ql−1 1 k π(k|l)qik=−11EvuRRi0((kk))
X X X u
and t 
we obtain
R
E i = p(y|δ(i)p(y|δ(0)))dy Dn(Rt)= π(l)Da(R(l))
R
r 0! Z l
p X
i−1
Dn(Rt+1)≤∆Xπ(i,k)2Xπ(l|i,k)qqk−−1101−Xπ(j|i,k,l)(1−βtDn(Rt))j−113 +O(Dn(Rt)2) (2)
i,k 4 l l @ j≥2 A5
We express D (R(l)) in terms of D (R ): Evolution of ∆ Evolution of Ω
a n t 0.7
∆∆nb 4
1−D (R(l)) ≤ 1−minD (R(l))≤1−β D (R ) 0.6 hyb 3.8
a a t n t 3.6
l 0.5 3.4
βas ≤soomninlaDsaβ(Rt(li)s).aTfhuunsc,twioenoobftatihneeqituearatitoionn(2s)u.ch that 00∆..34 Ω23..382 ΩΩnhbyb
Wte thenDcna(nRtp)rove that if Ω < 1 then there exists α 0.2 22..46
∆
such that if D (R ) < α at some iteration t , then 0.1 2.2
n t0 0 dv2
limt→∞Dn(Rt)=0. Moreover, if X(k) is a symmetric 0 816 32 64 12q8 256 81632 64 12q8 256
probability-vectorrandom variables of size q , then
k Fig. 3. Channel and code parameters ∆ and Ω for hybrid and non-hybrid
codesintermsofmaximumsymbolorderq.Thesefiguresshowthatahybrid
1
D (X(k))2 ≤P (X(k))≤(q −1)D (X(k)) (3) codecanbestablewhenanon-binary codeisnot.
q2 n e k n
k
Let us remind that Dn(Rt) = kπ(k)Dn(R(tk)), and hybrid stability condition reduces to non-hybridstability con-
thatthesequenceDn(Rt)∞t=t0 coPnvergestozero.Thusfor dition, given by:
allk, thesequencesD (R(k))∞ alsoconvergeto zero.
And hence P (R(k)) cnonvtergets=tt0o zero. This is true for Ωnb = ρ′(1)λ′(0)
e t q−1
all k, and since we have P (R ) = π(k)P (R(k)), 1 1
P (R ) also converges to ezerot. This pkroves thee stuffi- ∆nb = q−1 exp(−2σ2ni)
cieencyt of the stability condition. P Xi=1
with n , the number of ones in the binary map of i ∈ G(q).
i
(cid:3) Under this form, we can prove that ∆ tends to zero as q
nb
goes to infinity. On BI-AWGN channel, this means that any
Thus, we have proved that, provided that a fixed point of
fixed point of density evolution is stable as q tends to infinity
density evolution exists for hybrid codes, this point can be
fornon-binaryLDPCcodes,andforhybridcodestoo(because
stable under certain condition. Our hybrid codes have hence
of Remark 1).
threshold behavior.
Those results indicate that optimization procedures will be
more efficient since there exist more stable hybrid codes than
non-hybrid LDPC codes for a given set of channel and code
V. ANALYSIS OF THESTABILITY CONDITION
parameters.The optimizationandcode designis reportedin a
Now we are able to compare the stability conditions for future application.
hybridLDPCcodeswhosehighestordergroupisG(q)andfor
non-binary LDPC codes defined on the highest field GF(q). REFERENCES
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