The Velocity of the Decoding Wave for Spatially Coupled Codes on BMS Channels

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by  Rafah El-Khatib

The Velocity of the Decoding Wave for Spatially
Coupled Codes on BMS Channels
Rafah El-Khatib and Nicolas Macris
LTHC, EPFL, Lausanne, Switzerland
Emails: {rafah.el-khatib,nicolas.macris}@epfl.ch
Abstract—We consider the dynamics of belief propagation 0.4
decoding of spatially coupled Low-Density Parity-Check codes.
It has been conjectured that after a short transient phase, the 0.3
profile of “error probabilities” along the spatial direction of
a spatially coupled code develops a uniquely-shaped wavelike x(z)0.2
solution that propagates with constant velocity v. Under this
0.1
assumptionandfortransmissionovergeneralBinaryMemoryless
Symmetricchannels,wederiveaformulaforv.Wealsopropose
0
7 approximations that are simpler to compute and support our 0 10 20 30 40 50
1 findings using numerical data. z
0
2 Fig. 1. We consider the (3,6)-regular LDPC spatially coupled code with
I. INTRODUCTION L=50, w=3 on the BEC(0.46). We plot the error probability along the

n spatialdimensionandobservethe“decodingwave”.This“soliton”isplotted
a Spatial coupling is a construction of Low-Density Parity- every50iterationsuntiliteration250andisseentomakeaquicktransition
J Check (LDPC) codes that has been shown to be capacity- fromzeroerrorprobabilitytotheBP-valueoftheerrorprobability.

3 achieving on general Binary Memoryless Symmetric (BMS)
1 channels under Belief Propagation (BP) decoding [1]. The
capacity-achieving property is due to the “threshold satura- Asolitonicwavesolutionhasbeenproventoexistin[4]for
T] tion” of the BP threshold of the coupled system towards the the BEC, but the question of the independence of the shape
I maximuma-posteriori(MAP)thresholdoftheuncoupledcode from initial conditions is left open. In [4] and [5], bounds on
s. ensemble [1], [2]. the velocity of the wave for the BEC are proposed. In [6] a
c To study the performance of codes under spatial coupling, formulaforthewaveinthecontextofthecoupledCurie-Weiss
[
it is useful to analyze the decoding profile along the spatial toy model is derived and tested numerically.

1 axis of coupling. For the sake of the discussion let the integer In this work we derive a formula for the velocity of the
v z∈{0,...,L−1}denotethepositionalongthespatialdirection waveinthecontinuumlimitL(cid:29)w(cid:29)1,withtransmissionover
4 ofthegraphconstruction,whereListhelengthofthecoupling general BMS channels (see Equ. (7)). Our derivation rests on
6
7 chain.InthegeneralframeworkofBMSchannelsthedecoding the assumption that the soliton indeed appears. For simplicity
3 profileconsistsofthevectorofprobabilitydistributionsofthe welimitourselvestothecasewheretheunderlyinguncoupled
0 log-likelihoodsofbitsunderBPdecoding.Thez-thcomponent LDPC codes has only one nontrivial stable BP fixed point.
1. of this vector, call it xz, equals the log-likelihood distribution For the specific case of the BEC the formula greatly sim-
0 of the bits located at the z-th position. In the special case of plifiesbecausedensityevolutionreducestoaone-dimensional
7 the Binary Erasure Channel (BEC) this reduces to a vector of scalarsystemofequations.Forgeneralchannelswealsoapply
1 erasureprobabilities0<x <1.Thisdecodingprofilesatisfies theGaussianapproximation[7],whichreducestheproblemto
: z
v a set of coupled Density Evolution (DE) iterative equations. a one-dimensional scalar system and yields a more tractable
i It has been proven that under DE iterations, as long as the velocity formula. We compare the numerical predictions of
X
channel noise is below the MAP threshold, DE iterations thesevelocityformulaswiththeempiricalvalueofthevelocity
r
a drive xz to the all-∆∞ vector (the Dirac mass at infinite log- for finite L and w.
likelihood, i.e. perfect knowledge of the bits) [1], [2]. In the We propose a further approximation scheme (valid close to
specialcaseoftheBECthiscorrespondstoavectoroferasure theMAPthreshold)thatexpressesthevelocitysolelyinterms
probabilities driven to zero by DE iterations [3]. of degree distributions of the code (in the spirit of [4], [5]).
An interesting phenomenon that occurs during decoding, Thismaybeusefultoprovideeasy-to-handledesignprinciples
when the channel noise is between the BP and the MAP that maximize the velocity.
thresholds, is the appearance of a decoding wave or soliton. Ourformulacanbeappliedtoestimateparametersinvolved
It has been observed that after a transient phase the decoding in the scaling law [8] of finite-size ensembles. This is briefly
profile develops a fixed shape that seems independent of the discussed as a further possible application.
initial condition and travels at constant velocity v. The soliton The derivation of the velocity formula combines the use of
is depicted in Fig. 1 for the case of a (3,6)-regular spatially the “potential functional” introduced and used in a series of
coupled code on the BEC. Note that the same phenomenon works([2],[3],[9],[10])andthecontinuumlimitL(cid:29)w(cid:29)1
has been observed for general BMS channels. which makes the derivations analytically tractable.
In section II, we describe the setup and notation. In section The DE equation is obtained by setting to zero the functional
III, we state our main result and show a sketch of the its derivative ofW(x;c) with respect to x.
s
derivation.TheGaussianapproximation,theapplicationtothe The BP threshold h is strictly smaller than the MAP
BP
BEC as well as further approximations, and the application to threshold h . Spatial coupling, however, exhibits the attrac-
MAP
finite-sizeensemblesarediscussedinsectionIV.Moredetails tive property of threshold saturation which makes it possible
for some of the derivations can be found in [11]. to decode perfectly up till h . The definitions of the BP and
MAP
MAP thresholds above extend to the spatially coupled setting.
II. POTENTIALFORMULATIONANDCONTINUUMLIMIT
We consider (almost) the same setting and notation as in C. Spatially coupled system
[2]. One important difference is that we will later consider Since the natural setting for coupling is discrete, we first
thecontinuumlimit,whichisanapproximationofthediscrete describe the system in discrete space before taking the con-
expressionsintheregimeoflargespatiallengthLandwindow tinuum limit.
size w. The coupled LDPC(λ,ρ) code ensemble is defined as fol-
lows.ConsiderL+w“replicas”ofthesinglesystemdescribed
A. Preliminaries
in section II-B, on the spatial coordinates z∈{−w+1,...,L}.
Consider a symmetric probability measure x(α) on the
The system at position z is coupled to other systems by
extended real numbers R¯. These are measures satisfying
means of a uniform coupling window of width w. We de-
x(α)=eαx(−α) for all α ∈R¯. Here α ∈R¯ is interpreted (t)
note by x the check-node input distribution at position
z
as a “log-likelihood variable”. The (linear) entropy functional z∈{−w+1,...,L} on the spatial axis, and at time t∈N. We
is defined as
then write the DE equation of the coupled system in discrete
(cid:90)
H(x)= dx(α)log (1+e−α) (1) space as
2
(cid:32) (cid:33)
1 w−1 1 w−1 (cid:16) (cid:17)
and will play an important role. x(t+1)= ∑ c λ ∑ ρ x(t) . (4)
InthesequelwewillusetheDiracmasses∆0(α)and∆∞(α) z w i=0 z−i(cid:16) (cid:16) w j=0 (cid:24) z−i+j
at zero and infinite likelihood, respectively. We will also
Here c =c, for z∈{0,...,L} and c =∆ otherwise. We fix
need the standard variable-node and check-node convolution z z ∞
(t)
the left boundary to x =∆ for z∈{−w+1,...,−1}, for all
operators and for log-likelihood ratio (LLR) message z ∞
distribution(cid:16)s involv(cid:24)ed in DE equations (see [12]). t∈N. The initial condition on the right side is x(z0)=∆0, for
z∈{0,...,L}. The initialization to perfect information at the
B. Single system
left boundary is what allows seed propagation along the chain
Consider an LDPC(λ,ρ) code ensemble and transmission of coupled codes.
over the BMS channel. Here λ(y) = ∑lλlyl−1 and ρ(y) = We denote by x the profile vector. Then the expression of
∑rρryr−1 are the usual edge-perspective variable-node and the discrete potential functional is
check-node degree distributions. The node-perspective degree (cid:40)
distributions L and R are defined by L(cid:48)(y)=L(cid:48)(1)λ(y) and U(x;c)= ∑L 1 H(R (x ))+H(ρ (x )) (5)
R(cid:48)(y)=R(cid:48)(1)ρ(y). z=−w+1 R(cid:48)(1) (cid:24) z (cid:24) z
We denote by x(t)(α) the variable-node output distribution 1 (cid:16) (cid:16)1 w−1 (cid:17)(cid:17)(cid:41)
at time t∈N. We consider a family of BMS channels whose −H(x ρ (x ))− H c L ∑ ρ (x ) .
distributionc (α)inthelog-likelihooddomainisparametrized z(cid:24) (cid:24) z L(cid:48)(1) z(cid:16) (cid:16) w i=0 (cid:24) z+i
h
by the channel entropy H(ch)=h. On the BEC, for example, The coupled DE equation (4) is obtained by setting to zero
we have cε(α)=ε∆0(α)+(1−ε)∆∞(α), H(cε)=ε. the functional derivative of this potential with respect to x.
We can track the average behavior of the BP decoder by
D. Continuum Limit
means of the DE iterative equation [12]
Wenowconsiderthecoupledsysteminthecontinuumlimit
x(t+1)=c λ (ρ (x(t))) (2)
h (cid:16) (cid:24) (see[4],[13],[14]forthecaseoftheBEC)L→+∞andthen
(cid:16)
with initial condition x(0) =∆0. The BP threshold hBP is the w→+∞. We set x(wz,t)≡x(zt) and replace wz →z where the
largest value of h for which the DE recursion converges to new z is a continuous variable on the spatial axis, z∈R (we
∆ . slightly abuse notation here). The DE equation (4) becomes
∞
Fromnowonwewillomitthesubscripthandtheargument (cid:90) 1 (cid:16)(cid:90) 1 (cid:0) (cid:1)(cid:17)
α of the distribution c (α) to alleviate notation. Later on, x(z,t+1)= duc(z−u) λ dsρ x(z−u+s,t) ,
h (cid:16) (cid:24)
0 (cid:16) 0
c and c(z) describe the channel distribution at the spatial
z
position z in discrete and continuous settings, respectively. where c(z) is now the BMS channel distribution at the
The potential functionalW(x;c) (of the “single” or uncou- continuous spatial position z ∈ R (we again slightly abuse
s
pled system) is notation)andtheboundary/initialconditionsarex(z,t)=∆∞,
for z<0 and all t∈R / x(z,0)=∆ for z≥0.
1 0
W(x;c)= H(R (x))+H(ρ (x)) The potential functional W(x;c) of the coupled system in
s R(cid:48)(1) (cid:24) (cid:24)
the continuum limit is obtained from (5). In order to get a
1
−H(x ρ (x))− H(c L (ρ (x))). (3) finite result when L→+∞ we must normalize the potential
(cid:24) (cid:24) L(cid:48)(1) (cid:16) (cid:16) (cid:24) bysubtractingan“energy”associatedtoafixedprofilex that
0
where ∆E is the energy gap defined as ∆E =W(x ;c)−
s BP
0.015 Ws(∆∞;c). We recallWs(·;·) is the potential of the uncoupled
system (3), and ∆ the trivial fixed point (Dirac mass at infin-
∞
0.010 ity). With our normalizations Ws(∆∞;c)=0. Fig. 2 illustrates
theenergygap∆E forthe(3,4)-regularcodeensembleonthe
0.005 BEC(ε).
Formula (7) involves the soliton shape X. Using the DE
equationandx(z,t)=X(z−vt)wefindthatX(z)isthesolution
0.2 0.4 0.6 0.8 1.0
of
Fig. 2. The energy gap ∆E is the difference in the single potential at its
stablefixedpoints.Weplotthepotentialoftheuncoupled(3,4)-regularcode (cid:90) 1 (cid:16)(cid:90) 1 (cid:16) (cid:17)(cid:17)
ensembleatε=0.73.HereεBP=0.6473andεMAP=0.7456.Thegapalways X(z)−vX(cid:48)(z)= duc λ(cid:16) dsρ(cid:24) X(z−u+s) . (8)
vanisheswhenε=ε . 0 (cid:16) 0
MAP
Equs. (7)-(8) form a closed system of equations that can be
solved iteratively to obtain X and v.
satisfies x (z,t)→∆ when z→−∞ and x (z,t)→x when
0 ∞ 0 BP
z→+∞ (x is the nontrivial stable fixed point log-likelihood B. Brief Sketch of Derivation
BP
densityofBPforthesingleuncoupledcodeensemble),forall
Consider equation (6). Under the ansatz x(z,t)=X(z−vt)
t∈N. The functional is thus defined as follows,
forsmallvwegetx(z,t+1)−x(z,t)≈−vX(cid:48)(z−vt).Choosing
W =(cid:90) dz(cid:110) 1 (cid:16)H(R (x(z,t)))−H(R (x (z,t)))(cid:17) the direction η(z,t)=X(cid:48)(z−vt) we can rewrite (after a few
R R(cid:48)(1) (cid:24) (cid:24) 0 manipulations involving properties of ⊗ and (cid:1)) the left-hand
+H(ρ (x(z,t)))−H(x(z,t) ρ (x(z,t))) side of (6) as
(cid:24) (cid:24)
(cid:24)
−H(ρ (x (z,t)))+H(x (z,t) ρ (x (z,t))) (cid:90) (cid:16) (cid:17)
1 (cid:24) (cid:16)0 (cid:0)(cid:90) 10 (cid:24) (cid:24) 0 (cid:1)(cid:17) v RdzH ρ(cid:48)(cid:24)(X(z))(cid:24)X(cid:48)(z)(cid:24)2 .
− H c(z) L dsρ (x(z+s,t))
L(cid:48)(1) (cid:16) (cid:16) 0 (cid:24) We now consider the right-hand side of (6), which is the
1 (cid:16) (cid:0)(cid:90) 1 (cid:1)(cid:17)(cid:111) functional derivative of W(x;c) in the direction of η(z,t)=
+ H c(z) L dsρ (x (z+s,t)) .
L(cid:48)(1) (cid:16) (cid:16) 0 (cid:24) 0 X(cid:48)(z−vt). We first split W into two parts: the single system
potential W (x;c) that remains if we ignore the coupling
s
It can be shown that the integral converges under suitable
effect,andtherestwhichconstitutesthe“interactionpotential”
assumptions on the entropy of x as z→±∞.
W(x;c) (see [13] for similar splittings or [11] for the exact
Once the functional derivative of W in the direction η is i
definitions). With some care one can then show that
computed, one finds that the DE equation is equivalent to
δW δW
(cid:90) dzH(cid:16)(x(z,t+1)−x(z,t)) [ρ(cid:48)(cid:24)(x(z,t)) η(z,t)](cid:17) δxs[X(cid:48)(z)]=Ws(xBP;c)−Ws(∆∞;c), δxi[X(cid:48)(z)]=0.
R (cid:16) (cid:24)
δW We conclude that the “interaction” part does not contribute to
= [η(z,t)] (6)
δx the velocity and only the energy gap remains.
This is a gradient descent equation in an infinite-dimensional
IV. APPLICATIONS
space of measures.
A. Gaussian Approximation
III. MAINRESULT
In order to simplify the analysis of a spatially coupled
A. Velocity Formula ensemble with transmission over general BMS channels, one
We restrict ourselves to code ensembles with a single mayusethemethodofGaussianapproximations[7].Theidea
nontrivial stable BP fixed point. Consider the case when is to assume that the densities of the LLR messages appear-
the channel entropy h satisfies h <h<h . After a few ing in the DE equations are symmetri√c Gaussian densities
iterations of DE, which we callBtPhe “transiMeAnPt phase”, one (symmetric Gaussian densities x(α)=( 2πσ)−1exp(−(α−
observes a solitonic (wavelike) behavior as depicted in Figure m)2/2σ2)arecharacterizedbytherelationσ2=2m).Further-
1. This motivates us to make the following assumptions: (i) more the channel density c is replaced by a BIAWGNC(σn2)
after a transient phase the profile develops a fixed shape X(·); with the same entropy H(c).
(ii) the shape is independent of the initial condition; (iii) the Thesystembecomesscalarone-dimensionalsinceoneonly
shape travels at constant speed v; (iv) the shape satisfies the trackstheevolutionofthemeansorequivalentlytheentropies
boundary conditions X(z)→∆ for z→−∞ and X(z)→x of the densities. For simplicity, we consider the ((cid:96),r)-regular
∞ BP
for z→+∞. We thus make the ansatz x(z,t)=X(z−vt). ensemble.Densityevolutionisconvenientlyexpressedinterms
Fromthisansatzandequation(6)weobtainourmainresult. of the entropies p(z,t) = H(x(z,t)). One then observes a
The velocity of the soliton is (primes are derivatives) “scalar” wave propagation much like the one of Fig. 1. The
entropy of a symmetric Gaussian density of mean m equals
∆E
v= (cid:82)RdzH(cid:16)ρ(cid:48)(cid:24)(X(z))(cid:24)(X(cid:48)(z))(cid:24)2(cid:17), (7) ψ(m)=(cid:0)√4πm(cid:1)−1(cid:90)Rdze−(z−m)2/4mlog2(1+e−z).
Withthisfunction,theGaussianapproximationforthevelocity
0.05
reads (primes are derivatives) 0.045 Analytical velocity
0.04 Empirical velocity
Approximation
WGA(p ;c)−W GA(0;c) 0.035
v = s BP s , (9) 0.03 Upper bound
GA −(r−1)(cid:82)Rdz(p(cid:48)(z))2ψ((cid:48)(cid:48)ψ(((cid:48)r(−ψ2−)1ψ(1−−1(p1(−z)p))()z2))) Velocity0.00.2052
0.015
where p(z)denotestheshapetheentropyprofile, p =H(x ),
BP BP 0.01
and (3) now becomes 0.005
0
0.465 0.47 0.475 0.48 0.485
WGA(p;c)=(cid:0)1−1(cid:1)ψ(cid:0)rψ−1(1−p)(cid:1)−ψ(cid:0)(r−1)ψ−1(1−p)(cid:1) Channel parameter epsilon
s r
+1−1ψ(cid:0)ψ−1(H(c))+(cid:96)ψ−1(cid:0)1−ψ((r−1)ψ−1(1−p))(cid:1)(cid:1) Foridge.r3.of tWheelepgloetndth)eofnothrmeadleizceoddivneglopcriotifielsevfBoErC0,.4v6e,3v<a2ε, a<ndεMvABP/=α0(.i4n88th1e,
r (cid:96) forthe(3,6)-regularcodeensembleforL=128,w=3.
(here ψ−1(H(c)) is the mean of the BIAWGNC(σ ) that has
n
thesameentropyasthechannelcandequals2/σ2).Theshape C. Further approximations for scalar systems
n
p(z) is computed from
The one-dimensional formulas for the velocity might help
design degree distributions. However it is costly to compute
(cid:90) 1 (cid:16)
p(z)−v p(cid:48)(z)= duψ ψ−1(H(c)) theshapeoftheprofile(thatentersthedenominators)forevery
GA
0 degreedistribution.Weproposeahierarchyofapproximations
+((cid:96)−1)ψ−1(cid:16)1−(cid:90) 1dsψ(cid:0)(r−1)ψ−1(1−p(z−u+s))(cid:1)(cid:17)(cid:17). for the velocity that involve only the degree distributions and
0 quantities related to the single system, i.e, no profile shape
needstobecomputed.Thesearegoodforε closetoε .The
B. Velocity on the BEC MAP
first two approximations of the hierarchy are, for i=1,2,
For the BEC we can directly simplify (7) (alternatively
W (x ;ε)−W (0;ε)
oapnperocxainmaretidoenrivoevetrhethfeormBuElCa).usFinogr dthirisectclayset,hethceoncthinaununmel vai = (cid:82)0xBPdxρBE(cid:48)C(1−BPx)(cid:112)−1B2ExC2+24Fi(x) (12)
dthisetrdibeuctoiodningiswcav=e iεs∆e0n+tire(1ly−chεa)∆ra∞c.terTizheed fibxyedthesh(aspcealaorf) wfh(eyr)e=Fyi(,x)f=(y)(cid:82)0=xdyy(−1v−ρ(cid:112)−−1(112−y2λ+−21(4fFi(y(y)/).ε)T)h)eadnedrivwahtieorne
erasure probability x(z), i.e., X(z) = x(z)∆0+(1−x(z))∆∞ is1not shown2 here due tao1 length constra1ints, but is to some
and X(cid:48)(z)=x(cid:48)(z)∆0−x(cid:48)(z)∆∞. Then using X(z) ∆0 =X(z), extent inspired from that of [6] for the coupled Curie-Weiss
X(z) ∆∞ =∆∞, X(z) ∆0 =∆0, X(z) ∆∞ =X(cid:16)(z), we find model introduced in [9], [10]. We note that the approximation
λ(cid:16)(X(cid:16)(z))→λ(x(z)),ρ(cid:24)(cid:24)(X(z))→1−ρ((cid:24)1−x(z)).Thevelocity scheme breaks down if the quantities under the square roots
becomes are negative; this depends on the code parameters.
One can also derive similar approximative formulas within
W (x ;ε)−W (0;ε)
vBEC= (cid:82) BdECzρ(cid:48)B(P1−x(z))BE(Cx(cid:48)(z))2, (10) the Gaussian approximation which also is a one-dimensional
R scalar system. This is not shown here.
where the single potential (3) now is
D. Numerical Simulations
1−R(1−x) εL(1−ρ(1−x)) In this section we compare numerical predictions with
W (x;ε)= −xρ(1−x)− .
BEC R(cid:48)(1) L(cid:48)(1) the velocity formulas for the cases of the BEC(ε) and
BIAWGNC(σ2), to the further approximations, and also to
n
Again, the erasure profile has to be computed from the one- theempiricalvelocity.Theempiricalvelocityv isthevelocity
e
dimensional equation calculated from the decoding profiles obtained by running DE
(4).Inparticular,itisequaltotheaverageof∆z/(w∆I),where
x(z)−v x(cid:48)(z)=ε(cid:90) 1duλ(cid:16)(cid:90) 1ds(1−ρ(1−x(z−u+s))(cid:17). ∆z is the spatial distance between the centers of the kinks (or
BEC 0 0 fronts) of two profiles, and ∆I is the number of iterations of
DE that were made to go from the first profile to the second.
The formula obtained here for the BEC is obviously very
It serves as the reference value of the velocity with which we
similar to the upper bound proved in [5] (Theorem 1) for a
compare our formula.
discrete system
Figure 3 shows velocities (normalized by w) for the BEC
for a (3,6) coupled code with L=128 and w=3 in a range
W (x ;ε)−W (0;ε)
v =α BEC BP BEC , α ≤2. (11) close to the MAP threshold. The velocity decreases to zero
B ∑ ρ(cid:48)(1−xz)(xz−xz−1)2 as ε approaches ε and confirms that the soliton becomes
z∈Z MAP
“static” at the MAP threshold (we point out that at ε the
MAP
In [5] it is conjectured based on numerical simulations that existence and unicity up to translations of the static shape
α =1 would be a tight bound. Our results here are perfectly has been proven by displacement convexity techniques [13],
consistent with the findings of [5]. [14]). We see that the theoretical velocity v and empirical
BEC
velocity v nicely match over a wide range of noise. The (by definition), then γ ≈rˆ | /∆ε. This parameter γ enters in
e 1 ε
approximation v works very well for ε close to ε , and the scaling law and is determined experimentally. Obviously
a2 MAP
also with respect to v with the added advantage that it does it would be desirable to have a theoretical handle on γ. It
B
not require running DE. Similar findings are also illustrated is argued in [8] that γ ≈γ¯ where γ¯=x v/∆ε and where v
BP
in table I for the coupled (3,6)-regular ensemble for L=256, is the velocity of the decoding wave. We find that for the
w=8, and for different values of ε. (3,6)ensemble,γ=2.155,γ¯=1.960;forthe(5,10)ensemble,
Table II compares the theoretical and empirical velocities, γ =2.095, γ¯ =1.733; for the (4,12) ensemble, γ =2.140,
v and v for a BIAWGNC(σ2). The velocities are obtained γ¯=1.778,tolistafewexamples.Thedifferencesmightmostly
GA e n
for the (3,6)- and the (4,8)-regular ensembles for different be related to the difference in ensembles considered here and
values of ψ−1(H(c))=2/σ2 (twice the signal to noise ratio). in [8], and also to the relatively large value of ∆ε = 0.04
n
(chosenin[8]duetostabilityissuesinnumericalsimulations).
TABLEI
NORMALIZEDVELOCITIESOFTHEWAVEONTHELDPC(x3,x6)ONTHE ACKNOWLEDGMENT
BEC(ε)WITHL=256FORw=8
R.E.thanksAndreiGiurgiuforfindingthebuginhercode.
We also thank Ru¨diger Urbanke for discussions.
ε 0.455 0.465 0.475 0.485
vBEC 0.05754 0.03741 0.02004 0.00456 REFERENCES
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Trans.Inform.Theory,vol.59,no.12,pp.7761–7813,2013.
[2] S. Kumar, A. J. Young, N. Macris, and H. D. Pfister, “Threshold
saturationforspatiallycoupledldpcandldgmcodesonbmschannels,”
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TABLEII
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2/σ2 2.33 2.35 2.38 2.40
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v ,(3,6) 0.0183 0.0222 0.0283 0.0325
GA [5] V.Aref,L.Schmalen,andS.tenBrink,“Ontheconvergencespeedof
ve,(3,6) 0.0183 0.0233 0.0317 0.0375 spatiallycoupledldpcensembles,”in51stAnnualAllertonConference
v ,(4,8) 0.0237 0.0258 0.0312 0.0381
GA onComm.,Control,andComp. IEEE,2013,pp.342–349.
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derived scaling law depends on “scaling parameters”, one of
[9] S. H. Hassani, N. Macris, and R. Urbanke, “Chains of Mean Field
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1
pp.2321–2325.
(around this mean) at any time during the decoding process.
As shown in [8] there exists a “steady state phase” where the
mean and the variance are constant, and during which one
can observe the appearance of the decoding wave. (Note that
here we consider one-sided termination instead of two-sided
termination in [8], so the fraction rˆ here is equal to half the
1
fraction called rˆ (∗) in [8]).
1
ConsidertransmissionovertheBEC(ε)andletε denote
((cid:96),r,L)
theBPthresholdofthefinite-sizeensemble.Wewritethefirst-
(cid:12) (cid:12)
or(cid:12)der Taylor expansion of rˆ1(cid:12)ε around ε((cid:96),r,L) >ε (cid:12)as rˆ1(cid:12)ε ≈
rˆ1(cid:12)ε((cid:96),r,L)+γ∆ε where∆ε=ε((cid:96),r,L)−ε.Thus,sincerˆ1(cid:12)ε((cid:96),r,L)=0