Spatially-Coupled Precoded Rateless Codes with Bounded Degree Achieve the Capacity of BEC under BP decoding

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by  Kosuke Sakata

Spatially-Coupled Precoded Rateless Codes with
Bounded Degree Achieve the Capacity of BEC
under BP decoding
Kosuke Sakata, Kenta Kasai and Kohichi Sakaniwa
Department of Communications and Computer Engineering,
Tokyo Institution of Technology
Email: {sakata,kenta,sakaniwa}@comm.ce.titech.ac.jp
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0
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Abstract—Raptor codes are known as precoded rateless codes decoding error probability steeply decreases with overhead
n
a thatachievethecapacityofBEC.Howeverthemaximumdegree α = 0 with bounded maximum degree over various BMS
ofRaptorcodesneedstobeunboundedtoachievethecapacity.In
J channels. However the decoding error probability was proved
thispaperweprovethatspatially-coupledprecodedratelesscodes
8 to be bounded away from 0 with bounded maximum degree
achieve the capacity with bounded degree under BP decoding.
2 for any α. This is explained from the fact that there are a
I. INTRODUCTION constant fraction of bit nodes of degree 0.
]
T Spatially-coupled (SC) low-density parity-check (LDPC) The authors presented empirical results in [8] showing that
I codes attract much attention due to their capacity-achieving SC MacKay-Neal (MN) codes and SC Hsu-Anastasopoulos
.
s performance under low-latency memory-efficient sliding- (HA) codes achieve the capacity of BEC with bounded max-
c
window belief propagation (BP) decoding. The studies on imum degree. Recently a proof for SC-MN codes are given
[
SC-LDPC codes date back to the invention of convolutional in [9] by using potential functions. It was observed that the

1 LDPC codes by Felstro¨m and Zigangirov [1]. Lentmaier et SC-MN codes and SC-HA codes have the BP threshold close
v
al.!observedthattheBPthresholdofregularSC-LDPCcodes to the Shannon limit in [10] over BMS channels.
8
8 coincides with the maximum a posterior (MAP) threshold of In[11],theauthorsproposedSCprecodedratelesscodesby
2 the underlying block LDPC codes with a lot of accuracy by concatenatingregularSC-LDPCcodesandregularSC-LDGM
7 density evolution [2]. Kudekar et al. proved that SC-LDPC codes. We derived a lower bound of the asymptotic overhead
1. codes achieve the MAP threshold of BEC [3] and the binary- from stability analysis for successful decoding by density
0 inputmemorylessoutput-symmetric(BMS)channels[4]under evolution. We observe that with a sufficiently large number
4 BP decoding. ofinformationbits, theasymptoticoverheadandthedecoding
1 Ratelesscodesareaclassoferasure-recoveringcodeswhich error rate approach 0 with bounded maximum degree.
:
v producelimitlesssequenceofencodedbitsfromkinformation In this paper we give a proof that the SC precoded rateless
Xi bits so that receivers can recover the k information bits from codes with possible smallest degree achieve the capacity of
arbitrary(1+α)k/(1−ǫ)receivedsymbolsfromBEC(ǫ). We BECunderBPdecoding.Theproofneedstousedualityprop-
r
a denoteoverheadbyα.Designingratelesscodeswithvanishing ertyofpotentialfunctions,sincethisavoidfacingdifficultiesto
overhead is desirable, which implies the codes achieve the proveaninequalitywithaparametricallydescribedconstraint.
capacity of BEC(ǫ). LT codes [5] and raptor codes [6] are
rateless codes that achieve vanishing overhead α → 0 in
II. PRELIMINARIES
the limit of large information size over the BEC. By a nice
analogybetweentheBECandthepacketerasurechannel(e.g.,
A. (d,d ,d ) Precoded-Rateless Code C
l r g
Internet), rateless codes have been successfully adopted by
several industry standards. Throughout this paper, we deal with a randomly con-
A raptor code can be viewed as concatenation of an outer structed code as a random variable. In this section, we
high-rate LDPC code and infinitely many single parity-check define a precoded-rateless code C. This induces spatially-
codesoflengthd,wheredischosenrandomlywithprobability coupled precoded-rateless code C. Let k be the number of
Ω ford≥1.Raptorcodesneedtohaveunboundedmaximum information bits. For d ≥ 2,d ≥ 3,d ≥ 2, we define
d l r g
degree d for Ω 6=0. This leads to a computation complexity (d,d ,d ) precoded-ratelesscodebas follows. Construction of
d l r g
problem both at encoders and decoders. (d,d ,d ) precoded-rateless code C involves two encoding
l r g
Aref and Urbanke [7] investigated the potential advantage steps:outercode(precode)andinnercode.Firstkinformation
of universal achieving-capacity property of SC low-density bits are encoded by (d,d ) regular LDPC codes into M bits
l r
generator matrix (LDGM) codes. They observed that the x(0),...,x(M−1).Thecodingrateof(d,d )regularLDPC
l r
codes R is given as follows. 3. f(g(0;ǫ);ǫ)=0, and F(g(0;ǫ);ǫ)=0.
pre
dl Definition 2 ([14, Def. 2]). Generally, the potential function
R =1− .
pre
dr U(x;ǫ) of a vector admissible system (f,g) is defined as
From this, we have k =R M. follows.
pre
We further pre-coded M into infinitely long code bits by
U(x;ǫ)=g(x;ǫ)DxT −G(x;ǫ)−F(g(x;ǫ);ǫ).
inner code. At each time t∈[1,∞), repeat 1 and 2.
1. Choose d indices l(t),...,l(t) ∈ [0,M −1] uniformly
g 1 dg For a code with d-variable dd (µ,ν) is given by
at random.
2. Calculate the sum of the following d bits and transmit µ 1−x(ℓ);1−ǫ
g y(ℓ) =1− j =: g x(ℓ);ǫ , (1)
the sum. j (cid:0) µj(1;1) (cid:1) j
h (cid:16) (cid:17)i
x(l1(t))+···+x(ld(tg)). x(ℓ+1) = νj y(ℓ);ǫ =: f y(ℓ);ǫ , (2)
Consider a receiver receives n symbols y(1),...,y(n) j ν(cid:0)j(1;1)(cid:1) h (cid:16) (cid:17)ij
through BEC(ǫ). We define the overhead as for i=1,...,d, where ν (x;ǫ), ∂ ν(x;ǫ) and µ (x;ǫ),
α= nk(1−ǫ)−1. a∂n∂xdjµth(xe;pǫo)t.eSnuticahlafusnycsttieomnj(isfg,giv)eins∂aasxvjfeocltloorwasdm[9i]ssibjlesystem
Zero overhead α = 0 i.e. k := (1 − ǫ) implies capacity
achieving over BEC(ǫ). n U(x;ǫ)=µ(1,1−ǫ)−µ(1−x;1−ǫ)−ν(g(x;ǫ);ǫ)
The factorgraph[12] fordecodingconsistsof M bitnodes d
and dlM parity-checknodesofpre-codeandnchannelfactor − xjµj(1−x;1−ǫ). (3)
dr
nodes1[x(l1(t))+···+x(ld(tg))=y(t)]fort=1,...,n.LetΛd Xj=1
be the probabilitythata bit nodeconnectsto d channelnodes Definition 3 ([14, Def. 7]). The potential threshold is
and let β be the expected number of channel nodes to which
a bit node connect to. For M →∞, we have ǫ ,sup{ǫ∈E |inf U(x;ǫ)>0}.
∗ x (ǫ)
∈F
d R (1+α)
g pre
β := , Thisis well definedonly if{ǫ∈E |inf U(x;ǫ)>0}=6
1−ǫ x (ǫ)
∅. Let ǫ be the uncoupled system thre∈shFold defined in [14,
βde β ∗s
Λ(x)= Λdxd =e−β(1−x) = d!− xd, Def. 6]. For ǫ∗s <ǫ<ǫ∗, the energy gap,
d 0 d 0
X≥k d d X≥d ∆E(ǫ), max inf U(x;ǫ′),
R := = gR = g 1− l , ǫ′ [ǫ,1]x (ǫ′)
total n β pre β d ∈ ∈F
r
(cid:16) (cid:17) is well defined and strictly positive.
where R is equal to the coding rate. Assume the trans-
total
mission takes place over BEC(ǫ), Shannon limit is given by Definition4([14,Def.9]). Wedefinespatially-coupledsystem
ǫSha :=1−R =1− dg 1− dl . {x(ℓ)} of coupling numberL and couplingwidth w is
The multivtaortiaalte degreβe distridbrution (dd) [13], [9] of defiinedi≥a0s,ℓ≥0
(cid:16) (cid:17)
(d,d ,d ) precoded-rateless code is given by
l r g x(0) =1,
d d d i
ν(x;ǫ)= βgxd1lΛ(x2), µ(x;ǫ)= βgdrlxd1r +ǫxd2g. x(t+1) = 1 w−1f 1 w−1g(x(t) ;ǫ );ǫ ,(4)
B. Vector Admissible System i w w i+j−k i+j−k i−k!
k=0 j=0
X X
Definition 1. ForX ,[0,1]d, andfunctionsF :X×[0,1]→ ǫ i∈{0,...,L−1}
R, G : X ×[0,1] → R satisfying G(0,ǫ) = 0 and a d×d ǫi =
(0 i∈/ {0,...,L−1}.
positive diagonal matrix D, consider the following recursion
x(0) =1, If we take (f,g) as (1) and (2), SC-system {xi(ℓ)}i 0,ℓ 0
≥ ≥
x(ℓ+1) =f(g(x(ℓ);ǫ);ǫ), gives the DE of spatially-coupled code with dd (µ,ν).
where we define f : X ×[0,1] → X, g : X ×[0,1] → R, Theorem 1 ([14, Thm. 1]). For a vector admissible sys-
tem (f,g) with potentialthresholdǫ . Thereexists a constant
F (x;ǫ) = f(x;ǫ)D and G(x;ǫ) = g(x;ǫ)D. We say that ∗
′ ′
the system x(ℓ) or equivalently (f,g) is a vector admissible Kf,g such that if ǫ < ǫ∗ and w > (dKf,g)/(2∆E(ǫ)), the
ℓ 0 only fixed point of the SC-system of coupling width w is 0.
system if ≥
1. f,g are twice continuously differentiable, From this theorem, it follows that a potential threshold of
2. f(x;ǫ),g(x;ǫ) are non-decreasing (w.r.t. (cid:22)) in x and DE system (1) and (2), can be achieved on SC-system in (4)
ǫ, by BP decoding.
C. Dual Code A. Potential Function of C at Fixed Points
⊥
For a code C with dd (ν,µ), we define the dualcode C⊥ as From (3) and (6), it follows that the DE system of C⊥ is
a code with dd (ν ,µ ) = (µ,ν). The DE of the dual code given by
⊥ ⊥
is given [9] by x(ℓ+1) =f (g (x(ℓ);ǫ);ǫ), where
⊥ ⊥ x(1ℓ+1) =(1−(1−x(1ℓ))d⊥r −1Λ(1−x(2ℓ)))d⊥l −1, (7)
f⊥(x,ǫ),1−g(1−x;1−ǫ), x(2ℓ+1) =ǫ(1−(1−x(1ℓ))d⊥r Λ(1−x(2ℓ)))d⊥g−1.
g (y,ǫ),1−f(1−y;1−ǫ).
⊥
The potential function of the DE system is given as follows.
Next lemma gives duality property over C and C⊥ [9]. U⊥(x1,x2,ǫ)= d⊥g − d⊥g (1−x1)d⊥r Λ(1−x2)
Lemma 1 ([9, Lem. 2]). Let C be a code with dd (ν,µ). Let β β
f(x⊥⊥(,gǫ⊥)(bxe⊥a;ǫ)fi)x.eTdhpeoni,n(txo,f1t−heǫD) Eis oaffiCx⊥ed. IpnoipnrteocfisDe,Exo⊥f C=, − dβ⊥gdd⊥r x1d⊥ld⊥r−1 +ǫ(cid:16)(1−(1−x1)(1−x1d⊥l1−1)(cid:17))d⊥g
where x,f(1−x⊥;1−ǫ). Let the potential functions of C (cid:16) ⊥l (cid:17)
d d
and C⊥ be U and U⊥, respectively. Then it holds that − ⊥g ⊥r x1(1−x1)d⊥r −1Λ(1−x2)
β
U⊥(x⊥;ǫ)=U(f(1−x⊥;1−ǫ);1−ǫ) +(cid:16)d⊥g(1−x1)d⊥r x2Λ(1−x2) .
+ν(1,1−ǫ)−µ(1,ǫ).
(cid:17)
We call (0,0;ǫ) and (1,ǫ;ǫ) trivial fixed points of the DE
From this lemma, by switching C and C⊥, it follows that system (7)Other fixedpointscan be written parametricallyas
(x⊥,1−ǫ)isafixedpointofDEofC⊥,wherex⊥ ,f⊥(1− x,x⊥2(x);ǫ⊥(x) with x∈(0,1).
x;1−ǫ) and
(cid:0) (cid:1) 1
U(x;ǫ)=U⊥(f+⊥µ(1(1−,1x−;1ǫ−)−ǫ)ν;1(1−,ǫǫ)). (5) x⊥2(x)=−β1 ln(11−−xx)dd⊥l⊥r−−11,
x (x)
2
ǫ (x)= .
III. PROOF OF CAPACITY ACHIEVABILITY ⊥ (1−(1−x)d⊥r Λ(1−x2(x)))d⊥g−1
We call these fixed points non-trivial.
In this section, we prove that the SC (d,d ,d ) precoded-
l r g
rateless code C achieves the capacity of BEC under BP
B. Potential Function of C at Fixed Points
decodingforsomebounded(d,d ,d ).Itissufficienttoshow
l r g
that the potentbial function of DE system of C is equal to The DE system of C is given as follows.
ShannonthresholdofC.Thestrategyoftheproofisasfollows.
WefirstprovethatthepotentialthresholdofDEsystemofC⊥ x(1ℓ+1) =(1−(1−x1)dr−1)dl−1Λ(1−(1−ǫ)(1−x2)dg−1),
isequaltoShannonthresholdofC⊥.Thenweprove,byusing x(2ℓ+1) =(1−(1−x1)dr−1)dlΛ(1−(1−ǫ)(1−x2)dg−1).
Lemma 1, that the potential threshold of DE system of C is
equal to Shannon threshold of C. The potential function of the DE system is given as follows.
Defined =d ,d =d,d =d .Theddofthedualcode
⊥l r ⊥r l ⊥g g d d
of C is given as follows. U(x ,x ,ǫ)= g l +1−ǫ
1 2
βd
r
ν⊥(x;ǫ)= dβ⊥gdd⊥r xd1⊥l +ǫxd2⊥g, − dβgddrl(1−x1)dr +(1−ǫ)(1−x2)dg
⊥l (6) (cid:16)d (cid:17)
µ⊥(x;ǫ)= dβ⊥g xd1⊥r Λ(x2). − βg(1−(1−x1)dr−1)dlΛ(1−(1−ǫ)(1−x2)dg−1)
d d
−{ g lx1(1−x1)dr−1+dg(1−ǫ)x2(1−x2)dg−1 .
β
Discussion1. Somereadersmightthinkthatitismorenatural
(cid:17)
to prove directly that the potential threshold of C is equal
We call (0,0;ǫ) and (1,1;ǫ) trivial fixed point of the DE
to Shannon threshold of C. If we choose that proof, we
system. Other fixed points can be written parametrically as
have to prove an inequality with a parametrically described
x,x (x);ǫ(x) with x∈(0,1).
2
constraint. The proof via C avoids such a difficult problem.
⊥
Furthermore, some readers might think that why we do not (cid:0) x(cid:1)2(x)=x(1−(1−x)dr−1),
cuosedes.pUantifaolrlytu-cnoautepllyedandduianltecroedsteinCgl⊥y,, Ci.e.,caCn⊥naost bae ruasteedleasss ǫ(x)=1+ ln(1−(1−x)xdr−1)dl−1.
a rateless codessince the innercodecan⊥noctbe viewed as an β(1−x2(x))dg−1
LDGM code. c We call these fixed points non-trivial.
TABLEI
C. Spatially-CoupledPrecoded-RatelessCode C Achievesthe
THESINGsgn[ψi(z)]OFSTURMSEQUENCE{ψi(z)}OFψ(z).
Capacity
i 0 1 2 3 4 5 6 7 8 9
In this section, we prove that SC (d,d ,db ) precoded- z=0 + 0 − − + + + − − +
l r g
z=1 + − − − − + + − − +
rateless code C achieves the capacity with (d = 2,d =
l r
3,d = 3) and (d = 3,d = 4,d = 3). First we claim
g l r g
the following thbeorem.
to the second term of (10), we have that (10) is greater than
Theorem 2. If the potential function of the DE system of
C⊥ is positive at any non-trivial fixed point (x⊥;ǫ), i.e., 1z2(3+6z−14z2+10z3−4z4)=: 1z2φ(z).
U (x ;ǫ)>0, thenthe potentialthresholdofthe DE system 6 6
⊥ ⊥
of C is equal to the Shannon limit of C. Since φ (z) = −4(7−15z +12z2) < 0 for z ∈ (0,1)and
′′
φ(0) = 3 > 0,φ(1) = 1 > 0, φ(z) > 0 for z ∈ (0,1). This
Proof:Recallthedefinitionofpotentialfunction.Itissufficient
concludes the lemma.
to show that
The SC (d = 2,d = 3,d = 3) precoded-rateless code C
l r g
U(1;ǫ)=ǫSh−a ǫ, (8) has many bit nodes of degree two. This leads to high error
U(x(x );ǫ(x ))>ǫSha−ǫ(x ). (9) floors. Next, let us consider SC (dl = 3,dr = 4,dg = 3b)
1 1 1
precoded-rateless code. This has no bit nodes of degree two.
Equation (8) is obvious from
Lemma 3. Let C be the dual code of the (d = 4,d =
⊥ l r
U(1;ǫ)=U⊥(0;1−ǫ)+µ(1,1−ǫ)−ν(1;ǫ)=ǫSha−ǫ 3,dg = 3) precoded-rateless code. Let U⊥ be the poten-
tial function of C . Then, for any non-trivial fixed point
⊥
followed from (5). Next, we will show (9). Let x;ǫ :=
(x,x (x),ǫ (x)), it holds that
⊥2 ⊥
(x ,x ;ǫ) be a fixed point of C. We see that x :=
1 2 ⊥
(x⊥1,x⊥2):=f⊥(1−x;1−ǫ(x1)),1−ǫ isafixedpo(cid:0)i(cid:0)ntof(cid:1)C⊥. U⊥(x,x⊥2(x1),ǫ⊥(x1))>0 for x∈(0,1).
Thismapstrivialandnon-trivialfixedpointsofC totrivialand
non-trivial fixed points of C , respec(cid:1)tively. Therefore, from Proof:ToshowU⊥(x,x⊥2(x),ǫ⊥(x))>0,itissufficientto
⊥ show that for z ∈(0,1),
the assumption it follows that U (f (1−x;1−ǫ(x ));1−
⊥ ⊥ 1
ǫ(x1))>0. By using (5) again, we have β U⊥(x,x⊥2(x),ǫ⊥(x)) z=x1/3 (11)
3z 1−z
U(x;ǫ(x1))=U⊥(f⊥(1−x;1−ǫ(x1));1−ǫ(x1)) = 4 (4−8z2+5z3)+(cid:12)(cid:12) (3−2z−2z3+2z4)log (−1+z3)2
+µ(1,1−ǫ(x1))−ν(1,ǫ(x1)) (cid:16) (cid:17)
By using 3−2z−2z3+2z4 >0 and giving upper bounds
>µ(1,1−ǫ(x ))−ν(1,ǫ(x ))
1 1
=1− dg(1− dl)−ǫ(x )=ǫSha−ǫ(x ). log(1+z+z2)≤z+ z2 − 2z3 + z4 + z5
β d 1 1 2 3 4 5
r
z2 z3
log(1−z)≤−z− −
2 3
The SC (d = 2,d = 3,d = 3) precoded-rateless code
l r g
for z ∈ (0,1) to each term in (11), we have (11) is greater
C has the smallest possible degree that satisfies the necessary
than
condition of capacity achievability [11].
1
Lbemma 2. Let C be the dual code of the (d = 2,d = z2(30+55z2−72z3−212z4+260z5−12z6−48z7)
⊥ l r 15
3,dg = 3) precoded-rateless code. Let U⊥ be the poten- 1
tial function of C . Then, for any non-trivial fixed point =: z2ψ(z).
⊥ 15
(x,x (x),ǫ (x)), it holds that
⊥2 ⊥ Let ψ (z), (i = 0,...,9) be Sturm sequence [15, p. 264]of
i
U⊥(x,x⊥2(x1),ǫ⊥(x1))>0 for x∈(0,1). a polynomial ψ(z). The number of sign changes of Sturm
sequence at z =0,1 are both 4. From this, it follows that the
Proof:ToshowU⊥(x,x⊥2(x),ǫ⊥(x))>0,itissufficientto number of roots of ψ(z) = 0 is zero. In Table I, we listed
show that for z ∈(0,1), the sign of Sturm sequence at at z = 0,1. Since ψ(0) =
30,ψ(1)=1, it follows that ψ(z)>0 for z ∈(0,1).
β U (x,x (x),ǫ (x)) (10)
⊥ ⊥2 ⊥ z=√x
=:z(3−3z+z2)+(−(cid:12)3+2z+2z2−2z3)log(1+z)>0. Theorem 3. For sufficiently large w and L, the spatially-
(cid:12) coupled (d,d ,d ) precoded-rateless code C with coupling
l r g
numberLandcouplingwidthw achievesthecapacityofBEC
Using −3+2z+2z2−2z3 <0 for z ∈(0,1) and giving an for (d,d ,d )=(2,3,3),(3,4,3). b
l r g
upper bound
Proof: The coding rate of C with fixed coupling width w
z2 z3 convergesto R in the limit of large L. Use Lemma 2 and
log(1+z)<z− + , z ∈(0,1) total
2 3 Lemma3 as the conditionof Tbheorem2, then we see thatthe
potential threshold of (dl,dr,dg) precoded-rateless code C is [12] F. Kschischang, B. Frey, and H.-A. Loeliger, “Factor graphs and the
equal to Shannon limit ǫSha :=1−R =1− dg 1− dl . sum-product algorithm,” IEEE Trans. Inf. Theory, vol. 47, no. 2, pp.
total β dr 498–519,Feb.2001.
Apply this result to Theorem 1, then we see that(cid:16)DE of(cid:17)C [13] T. Richardson and R. Urbanke, “Multi-edge type LDPC codes,”
converges to 0 if ǫ<ǫSha. 2003, preprint available at http://citeseerx.ist.psu.edu/viewdoc/
summary?doi=10.1.1.106.7310.
b [14] A. Yedla, Y.-Y. Jian, P. Nguyen, and H. Pfister, “A simple proof of
Discussion 2. In [11], we derived a necessary condition that
threshold saturation for coupled scalar recursions,” in Proc. 7th Int.
C achieves the capacity of BEC(ǫ) in the limit of w and L as Symp.onTurboCodes andRelated Topics,Sept.2012,pp.51–55.
follows. [15] W.Gautschi,NumericalAnalysis,ser.SpringerLink:Bu¨cher. Springer
Science+Business Media, LLC,2011.
b
d ln(d −1)
r r
d ≥
g
d −2
r
This is satisfied by (d,d ,d ) = (2,3,3),(3,4,3). We ob-
l r g
served thatfor manypatternsof(d,d ,d ) with no exception
l r g
thepotentialfunctionofC atnon-trivialfixedpointsispositive.
Fromthisobservation,webelievethatthenecessarycondition
is actually also a sufficiebnt condition for achieving capacity.
IV. CONCLUSION AND FUTURE WORK
We proved that SC precoded-rateless codes achieve the
capacityofBECwithboundeddegree.Theproofusedduality
propertyofpotentialfunctionsoverC andC ,sinceitiseasier
⊥
toprovethatpotentialthresholdisequaltoShannonthreshold
for C than C.
⊥
Futureworksincludeshowingsufficientconditionofcapac-
ityachievabilityonparameters(d,d ,d )andanextensionto
l r g
binary-input memoryless channels.
V. CONCLUSION
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