On the Error Probability of Short Concatenated Polar and Cyclic Codes with Interleaving

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by  Giacomo Ricciutelli

On the Error Probability of Short Concatenated
Polar and Cyclic Codes with Interleaving
Giacomo Ricciutelli, Marco Baldi, Franco Chiaraluce Gianluigi Liva
Università Politecnica delle Marche, German Aerospace Center (DLR),
Ancona, Italy Wessling, Germany
Email: [email protected],{m.baldi, f.chiaraluce}@univpm.it Email: [email protected]
7
1 Abstract—Inthispaper,theanalysisoftheperformanceofthe is carried out by introducing concatenated ensembles and by
0 concatenation of a short polar code with an outer binary linear deriving average weight enumerating function (AWEF) by
2 blockcodeisaddressedfromadistancespectrumviewpoint.The
following the well-known uniform interleaver approach [18].
analysis targets the case where an outer cyclic code is employed
n In the analysis, the knowledge of the input output weight
together with an inner systematic polar code. A concatenated
a code ensemble is introduced placing an interleaver at the input enumerating function (IOWEF) of the inner polar code is
J
of the polar encoder. The introduced ensemble allows deriving required.However,thepolarcodeIOWEFcalculationthrough
5
boundsontheachievableerrorratesundermaximumlikelihood analytical methods is still an unsolved problem. Hence, we
2
decoding, by applying the union bound to the (expurgated)
restrict our attention to short, high-rate polar codes for which
average weight enumerators. The analysis suggests the need of
] theproblemcanbesolvedthroughapragmaticapproach.More
T carefuloptimizationof theoutercode,toattainlow errorfloors.
precisely, we consider the dual code of the selected polar
I
. code and then we find its IOWEF by listing the codewords.
s I. Introduction
Subsequently, by using the generalized MacWilliams identity
c
[ Polar codes [1], [2] provably achieves the capacity of [19], we obtain the IOWEF of the original polar code. The

binary-inputdiscrete memoryless symmetric (BI-DMS) chan- weight enumerating function (WEF) of the outer cyclic code,
1
v nels by using the (low complexity) successive cancellation instead, is computed by following the method presented in
2 (SC) decoding algorithm, in the limit of infinite block length. [20].
6 At short block lengths polar codes under SC decoding tend
By adoptingthe uniform interleaver approach,we subsume
2 to exhibit a poor performance. In [3] it was suggested that
theexistenceofaninterleaverbetweentheinnerandtheouter
7
such a behavior might be due, on one hand, to an intrinsic
0 code, and obtain the average performance of an ensemble
weakness of polar codes and, on the other hand, to the sub-
. composed by the codes obtained by selecting all possible
1 optimality of SC decoding w.r.t. maximum likelihood (ML)
interleavers. Our analysis shows that the performance of the
0
decoding. An improved decoding algorithms were proposed
7 concatenatedschemewithandwithoutinterleaver(asproposed
1 in [3]–[5],while the structural propertiesof polar codes (e.g., in [3])maydiffer substantially.Similarly,byconsideringboth
: their distance properties) were studied, among others, in [6]– CRC and Bose-Chaudhuri-Hocquenghem(BCH) outer codes,
v
[11]. The minimum distance properties of polar codes can
i we show that the choice of the outer code plays an important
X be improved by resorting to concatenated schemes like the
role in the short block length regime.
r one of [3], where the concatenation of polar codes with an
a The paper is organizedas follows. Notation and definitions
outercyclicredundancycheck(CRC)codeisconsidered.This
areintroducedinSectionII.Thedistancespectrumanalysisof
solution, together with the use of the list decoding algorithm
theconcatenatedschemeisdiscussedinSectionIII.Numerical
of[3],allowsshortpolarcodestobecomecompetitiveagainst
resultsarereportedinSectionIV.Finally,SectionVconcludes
other families of codes [12]–[16].1
the paper.
A theoretical characterization of the performance of con-
catenated CRC-polar codes is still an open problem. Further-
more, for a fixed code length, a concatenated scheme can be
II. NotationandDefinitions
realized with several combinations of the component codes
parameters (e.g., one may choose various CRC polynomials
and polar codes designed for different target signal-to-noise Let w (·) be the Hamming weight of a vector. We denote
H
ratios). as N and K the outer cyclic code length and dimension,
In this paper, we provide an analysis of the concatenation respectively, while n and k identify the same parameters for
of polar codes with binary cyclic outer codes. The analysis the inner polar code. Therefore, the code rates of the outer
and inner code are R = K and R = k, respectively. The two
1Theon-going3GPPstandardization groupisconsidering theadoption of O N I n
codes can be serially concatenated on condition that N =k,
short polar codes with an outer CRC for the uplink control channel of the
upcoming5thgeneration mobilestandard [17]. thus the overall code rate of the concatenated code is R= K.
n
Given a binary linear code C(n,k), its WEF is defined as [19] (BEC) with erasure probability ǫ can be upper bounded as
[23]
N
AC(X)= AiXi (1) E[P (C,ǫ)]≤P(s)(n,k,ǫ)
Xi=0 B B
k n e e A¯
where Ai is the number of codewords c with wH(c)=i. In + ǫe(1−ǫ)n−emin1, ω (7)
this work we focus on systematic polar codes (the reason of Xe=1 e!  ωX=1 n! ωn 
tchoidsewchooridcecwcoililncbiededwisictuhssthede ilnafteorr)msaotiotnheveficrtsotrku,byitiseldoifnga whereA¯ω=E[Aω(C)]istheaveragemultiplici(cid:16)ty(cid:17)ofcodewords
c=(u|p)withpbeingtheparityvector.TheIOWEFofacode with wH(c) = ω and P(Bs) represents the BEP of an ideal
C(n,k) is maximum distance separable (MDS) code, with parameters
k n n and k. The union bound (UB) in (7) is applicable to every
AIO(X,Y)= AIOXiYω (2) code ensemble whose expected WEF is known. In order to
C i,ω
Xi=0ωX=0 use (7), the AWEF of the concatenated code ensemble must
be known.
where AIO is the multiplicity of codewords c with w (u)=
i,ω H When dealing with concatenated codes, it is commonplace
i and w (c)=ω. The enumeration of the codeword weights
H
to consider a general setting including an interleaver between
entails a large complexity even for small code dimensions.
theinnerandtheoutercodes.Theconcatenatedcodeensemble
In order to overcome this problem and obtain the IOWEF of
is hence given by the codes obtained by selecting all possible
the consideredpolar codes,we focuson short, high-ratepolar
interleavers.Thespecialcase withoutanyinterleavercanthen
codes and we exploit the generalized MacWilliams identity
be modeled as an identity interleaver. From [18], the AWEF
[19]. This approach was followed also for cyclic codes, for
ofaconcatenationformedbyaninnerpolarcodeandanouter
example,in[20]tocomputetheWEFofseveralCRCs.Denote
cyclic code can be obtained from the cyclic code WEF and
by C⊥ the dualcode of C. Given the dualcode WEF AC⊥(X), the polar code IOWEF as
we can express the original code WEF A (X) as [19]
C
AC(X)= (1|+C⊥X|)nAC⊥ 11−+XX! (3) A¯ω= N Aoiut·NAIi,Oω,in (8)
Xi=0 i
(cid:16) (cid:17)
where C⊥ is the cardinality of the dual code. When the where Aout is the weight enumerator of the outer code and
i
IOWEF(cid:12)(cid:12) is(cid:12)(cid:12)of interest, a significant reduction of the compu- AIO,in is the input-outputweightenumeratorof the inner code
tational(cid:12)cos(cid:12)tcanbeachievedbyconsideringsystematiccodes. i,ω
(we remind that the average multiplicities resulting from (8)
For such reason, in this work we have used only systematic
are, in general, real numbers).
inner polar codes. In the case of a systematic code C(n,k), it The ensemble C(n,k) contains the codes generated by all
is convenienttoderivethe IOWEFfromthe inputredundancy
possible interleavers. Thus, also bad codes (i.e., characterized
weight enumerating function (IRWEF) AIR(x,X,y,Y) defined
C by bad error rate performance) belong to the ensemble. It is
as
clear that the bad codes adversely affect the AWEF obtained
k r
AIR(x,X,y,Y)= AIRxk−iXiyr−pYp (4) through (8) causing a too pessimistic estimate of the error
C i,p
Xi=0Xp=0 probabilityobtainedthrough(7), with respectto thatachieved
by properly designed codes. A simple way to overcome this
where AIi,Rp is the multiplicity of codewords c with wH(u)=i issue is to divide C(n,k) into the bad and good code subsets,
andwH(p)=p,withwH(c)=i+pandn=k+r.Hence,starting and then derive the AWEF only of good codes through the
fromtheIRWEFofthedualcodeAIR(x,X,y,Y),wehave[21], expurgated C(n,k) [24]. In fact, A¯ =ξA¯g +(1−ξ)A¯b, where
C⊥ ω ω ω
[22] A¯g and A¯b denote the good and the bad codes ensemble,
ω ω
respectively.In this work we have assumed ξ=0.99, hence at
AIR(x,X,y,Y)=
C leastonecodebelongstothegoodcodesensemble.Therefore,
1
AIR(x+X,x−X,y+Y,y−Y). (5) when the expurgationmethod is adoptable (i.e., the first term
|C⊥| C⊥ of the AWEF has a codewords multiplicity less than ξ),
through(7)and(8)theaverageperformanceofthegoodcodes
Then, the IOWEF is obtained as
subset is derived.
AIO(X,Y)=AIR(1,XY,1,Y). (6) StudyingthedualcodesandexploitingMacWilliamsidenti-
C C
tiesallowconsiderablereductionsin complexityofexhaustive
III. UnionBoundoftheAverageBlockErrorProbabilityof
analysis as long as the original code rate is sufficiently large.
theConcatenatedScheme
Thiswillbethecaseforthecomponentcodesconsiderednext,
Given an ensemble of binary linear codes C(n,k), the which are characterized by RO,RI> 21. Therefore, in our case
expected block error probability (BEP) P of a random code (3) and (6) can effectively be exploited to calculate Aout and
B i
C∈C(n,k) under ML decodingover a binary erasure channel AIO,in in (8).
i,ω
IV. CodeExamples 100
Inthissectionwe considerseveralexamplesof polar-cyclic
10−1
concatenated codes and assess their performance through the
approach described in the previous sections. Our focus is on
short, high rate component codes, which allow to perform 10−2
seuxlhtasuasrteivoebatnaianlyedsiscoonfstihdeeirrindguaalsBiEnCarweaitshonearabsluerteimpero.bOaubrilriety- Probability10−3
iǫen.rrAloirtsektrranatonuwsreint,i,oiinnneptharceohbpaooblfailtrihtcyeofdiosellccooownnissnitdrgueecrextidao.mnTpahlfieesxreetfhdoervepa,loualeasroucfsotudhaeel Block Error 10−4
isdesignedbyusingǫ=0.3.Allperformancecurvesprovided 10−5
next are obtained through(7). We consider codes with n=64
bitsanda CRC ora BCH outercode,with N accordingto the 10−6 Polar code
Uniform interleaver
polar code dimension. In this work, we have used a CRC−8 No interleaver
Random interleavers
anda CRC−16with the followinggeneratorpolynomialsg(x) 10−7
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
ε
[20]:
• CRC−8: g(x)=x8+x2+1; Fig. 1. Estimated performance of concatenated codes with and without
• CRC−16−CCITT: g(x)=x16+x12+x5+1. interleaver composed by a (64,48) polar code and a CRC−8 code over the
BECunderMLdecoding.Performanceofthe(64,48)polarcodealoneisalso
Instead, regarding the BCH code, we study two BCH codes
reported.
that have the same redundancy as the CRC-8 and CRC-16
codes. Thus, forthe same R value,a performancecomparison
TABLEI
of the results achieved by using a CRC or a BCH outer code Numberofconcatenatedcodeswithrandominterleaverforagivenminimum
distancevalue
is feasible and fair. In order to keep the selected polar codes
unchanged, we have considered shortened BCH codes.
Concatenated scheme dmin=4 dmin=6 dmin=8
For the case without interleaver considered in [3], the
polar(64,48) +CRC-8 21 4 -
generator matrix G of the concatenated code can be obtained
polar(64,48) +BCH(48,40) 23 2 -
fromthecyclicandpolarcodegeneratormatricesGC andGP, polar(64,48)+CRC-16 1 8 16
respectively, as polar(64,48) +BCH(48,32) - - 25
polar(64,56)+CRC-16 16 9 -
G=G ·G . (9)
C P
When we instead consider the more general case with an
concatenatedschemesisnotcompletelyfairatleastbecauseof
interleaver between the inner and outer codes, we have
thedifferentcoderates;however,inthisway,theperformance
gain achieved through concatenation can be pointed out.
G=G ·I·G (10)
C P
Figures 1 and 2 show the UB of the concatenated scheme,
where I is an N×N permutation matrix representing the with and without random interleaver, formed by a polar code
interleaver. Considering all codes in C(n,k), (8) leads to their with R = 0.75 and an outer cyclic code with R = 0.83
I O
AWEFand henceto the averageperformancein termsof UB, (i.e., R = 0.625) in terms of BEP. In Fig. 1 and Fig. 2
according to the uniform interleaver approach. In order to a CRC-8 and a (48, 40) BCH outer code is considered,
have an idea of the gap between this average performance respectively, the latter obtained by shortening the (255, 247)
and those of single codes in the ensemble, in each of the fol- BCH code. In both the examples, the result obtained through
lowingfigureswe includethe performanceofcodesrandomly the AWEF corresponds to an average behavior, while some
picked in C(n,k). Withoutchangingthe outer and inner code, interleaver configurations achieve a smaller BEP. This trend
this can be easily done by introducing a random interleaver is due to the different minimum distance d of the codes.
min
betweenthecycliccodeandthepolarcodeintheconcatenated The results in Figs. 1 and 2 are very similar, showing that,
scheme. Hence, in this case, the matrix I in (10) is a random for this particular case, there is no substantial difference in
permutationmatrix. For readability reasons, in addition to the performancearisingfromthedifferenttypeofoutercode.This
solutionwithoutinterleaver,ineachofthefollowingexamples, conclusion is also supported by the results in Tab. I, where
wehaveconsideredonly25randominterleavers;however,the the number of concatenated schemes with random interleaver
obtained results allow to address some general conclusions. correspondingto a minimumdistance value is reported.From
Moreover, when the results of the expurgated C(n,k) differs these figureswe can observethe performancegain introduced
from that achievedby the AWEF, its performancein terms of by the concatenated scheme with respect to the (64,48) polar
UB is also considered. The UB of the considered polar code code used alone, that has d =4. In both the examples the
min
alone is also included as a reference. Clearly, the comparison AWEF and the concatenated code without interleaver have
between the performance of the polar code and those of the d =4 and d =6, respectively.
min min
100 100
10−1 10−1
10−2 10−2
Probability10−3 Probability10−3
Block Error 10−4 Block Error 10−4
10−5 10−5
Polar code
10−6 Polar code 10−6 Uniform interleaver
Uniform interleaver Uniform interleaver (Exp. AWEF)
No interleaver No interleaver
Random interleavers Random interleavers
10−7 10−7
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
ε ε
Fig. 2. Estimated performance of concatenated codes with and without Fig. 3. Estimated performance of concatenated codes with and without
interleaver composed by a(64,48) polar code and a (48,40)shortened BCH interleaver composed bya (64,48)polar code and a CRC−16code over the
code over the BEC under ML decoding. Performance of the (64,48) polar BECunderMLdecoding.Performanceofthe(64,48)polarcodealoneisalso
codealoneisalsoreported. reported.
100
In Fig. 3 and Fig. 4 the UB of the concatenated codes,
with and without interleaver, composed by a polar code with 10−1
R =0.75 and an outer code with R =0.66 (i.e., R=0.5)
I O
in terms of BEP is plotted. In Figs. 3 and 4 a CRC-16 10−2
and a (48, 32) BCH outer code is considered, respectively,
tDhieffelartetnetrlyobfrtaoimnedthebyprsehvoirotuesnifinggutrhees,(t2h5e5,U2B39o)btBaCinHedcwoditeh. Probability10−3
t2h,etheexpcuurrgvaeteodbtAaiWneEdFthisroungowh thaveaiAlaWblEeF. AwselilndFesigcrsi.be1satnhde Block Error 10−4
ensemble average performance, while we see that the curve
10−5
corresponding to the expurgated AWEF belongs to the group
Polar code
ofbestcodes.Alsointhesecasestheperformancegapbetween
10−6 Uniform interleaver
the (64,48) polar code alone and the concatenated codes is Uniform interleaver (Exp. AWEF)
No interleaver
remarkable. In both the examples the AWEF has d = 6, Random interleavers
min 10−7
while the expurgated AWEF and the concatenated scheme 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
ε
without interleaver have d = 8. However, from Fig. 4,
min
we can observe that, on the contrary to Fig. 3, all curves Fig. 4. Estimated performance of concatenated codes with and without
interleaver composed bya (64,48) polar code and a(48,32) shortened BCH
of concatenated codes are very close. In fact, for the case
code over the BEC under ML decoding. Performance of the (64,48) polar
in Fig. 4 only the AWEF results in dmin = 6 but with a codealone isalsoreported.
codewordsmultiplicityequalto0.0336,insteadtherealizations
of concatenated codes have d = 8, as shown in Tab. I.
min
Therefore, differently from the CRC, in this case the use of observe that, in this case, the introduction of a random inter-
a BCH code is able to increase the minimum distance of the leaver can improve the scheme without interleaving. Also for
concatenated code (also for the solution without interleaver); this example, Tab. I summarizes the number of concatenated
thus, for this specific case, the BCH code should be preferred schemeswith interleaverfor each value of the code minimum
to the CRC code. distance. In this case the polar code has d =2, while both
min
In all the previous figures the curve of the concatenated the AWEF and the concatenated scheme without interleaver
scheme without interleaver proposed in [3] falls within the have d =4. Instead, the expurgated AWEF belongs to the
min
group of best performingcodes. This may lead to the conclu- group of good codes with d =6. We have found a similar
min
sion that this configuration always produces a good result; in result also by using the CRC-8 code in place of the CRC-
reality this trend is not preserved for any choice of the code 16 but it is omitted for the sake of brevity. So, these counter
parameters. An example of the latter kind is shown in Fig. 5, examples (others can be found) clearly demonstrate that the
where a polar code with R =0.875 and a CRC-16 code (i.e., use of a selected interleaver may be beneficial from the error
I
R =0.71 and R=0.625) are considered. From the figure we rate viewpoint.
O
100 [8] M. Valipour and S. Yousefi, “On probabilistic weight distribution of
polar codes,” IEEE Commun. Lett., vol. 17, no. 11, pp. 2120–2123,
10−1 Nov.2013.
[9] Z. Liu, K. Chen, K. Niu, and Z. He, “Distance spectrum analysis of
polar codes,” in Proc. IEEE Wireless Commun. and Networking Conf.
10−2 (WCNC),Istanbul, Turkey,Apr.2014,pp.490–495.
[10] B. Li, H. Shen, and D. Tse, “A RM-polar codes,” CoRR, vol.
Probability10−3 [11] Moabf.sp/B1oa4lar0dr7e.ct5,o4dV8e.3sD,f2rra0og1mo4i.,a[AOn.neOwlintmep]oa.lnyAin,voaamnildaiabJll.efP:o.hrTmttiplal:li/ic/sahmr,x,“”iAv.ilongregPb/arroabicsc/.1pI4Er0oE7pE.e5r4tIi8ne3ts.
Block Error 10−4 [12] BSdye.mcLopdi.,eIrHnff.o.SrThpheoenloa,rryacn(odIdSeIDTs.)w,TiBsthea,rcc“yeAclolnincaa,rdeSadpputanividnea,nsJcuuycl.cce2hs0es1civk6e,,”pcIpaE.nEc2Ee3l0lCa–to2imo3n4m.ulinst.
10−5 Lett.,vol.16,no.12,pp.2044–2047, Dec.2012.
[13] K. Niu and K. Chen, “CRC-aided decoding of polar codes,” IEEE
Polar code Commun.Lett.,vol.16,no.10,pp.1668–1671, Oct.2012.
10−6 Uniform interleaver [14] A.Bhatia,V.Taranalli,P.H.Siegel,S.Dahandeh,A.R.Krishnan,P.Lee,
Uniform interleaver (Exp. AWEF)
D. Qin, M. Sharma, and T. Yeo, “Polar codes for magnetic recording
No interleaver
Random interleavers channels,”inProc.IEEEInf.TheoryWorkshop(ITW),Jerusalem,Israel,
10−7 Apr.2015,pp.1–5.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
ε [15] D.Wu,Y.Li,X.Guo,andY.Sun,“Orderedstatisticdecodingforshort
polarcodes,”IEEECommun.Lett.,vol.20,no.6,pp.1064–1067,Jun.
Fig. 5. Estimated performance of concatenated codes with and without 2016.
interleaver composed bya (64,56) polar code anda CRC−16 code over the [16] G. Liva, L. Gaudio, T. Ninacs, and T. Jerkovits, “Code design for
BECunderMLdecoding.Performanceofthe(64,56)polarcodealoneisalso short blocks: A survey,” CoRR, vol. abs/1610.00873, 2016. [Online].
reported. Available: http://arxiv.org/abs/1610.00873
[17] Huawei,“EvaluationofTBCCandpolarcodesforsmallblocklengths,”
3GPPTSGRANWG1N.85,Tech.Rep.,2016.
[18] S.BenedettoandG.Montorsi,“Unveilingturbocodes:someresultson
V. Conclusion parallelconcatenatedcodingschemes,”IEEETrans.Inf.Theory,vol.42,
no.2,pp.409–428,Mar.1996.
In this paper, the analysis of the performance of the con-
[19] F.J.MacWilliams andN. J.A.Sloane, The theory of error-correcting
catenation of a short polar code with an outer binary linear codes.IandII. North-Holland PublishingCo.,1977.
block code is addressed from a distance spectrum viewpoint. [20] J.K.WolfandR.D.Blakeney, “Anexactevaluation oftheprobability
of undetected error for certain shortened binary CRC codes,” in Proc.
The analysis is carried out for the case where an outer
21stIEEEMilitary Commun.Conf.,vol.1,SanDiego,CA,USA,Oct.
cyclic code is employed together with an inner systematic 1988,pp.287–292.
polar code. By introducing an interleaver at the input of the [21] C.Weiss,C.Bettstetter,andS.Riedel,“Codeconstructionanddecoding
of parallel concatenated tail-biting codes,” IEEE Trans. Inf. Theory,
polar encoder, we show that remarkable differences on the
vol.47,no.1,pp.366–386, Jan.2001.
block error probability at low erasure probabilities can be [22] F. Chiaraluce and R. Garello, “Extended Hamming product codes
observed for various permutations. The variations are due to analyticalperformanceevaluationforlowerrorrateapplications,”IEEE
Trans.onWirelessCommun.,vol.3,no.6,pp.2353–2361, Nov.2004.
thechangeintheoverallconcatenatedcodeminimumdistance
[23] G.Liva,E.Paolini,andM.Chiani,“Boundsontheerrorprobabilityof
(andminimumdistancemultiplicity)inducedby thechoiceof blockcodesovertheq-aryerasurechannel,”IEEETrans.onCommun.,
the interleaver. Bounds on the achievable error rates under vol.61,no.6,pp.2156–2165, Jun.2013.
[24] A. Bennatan and D. Burshtein, “On the application of LDPC codes
maximum likelihood decoding are obtained by applying the
to arbitrary discrete-memoryless channels,” IEEE Trans. Inf. Theory,
union bound to the (expurgated)average weight enumerators. vol.50,no.3,pp.417–438, Mar.2004.
The results point to the need of careful optimization of the
outer code, at least in the short block length regime, to attain
low error floors.
References
[1] E.Arikan, “Channel polarization: A method for constructing capacity-
achievingcodesforsymmetricbinary-inputmemorylesschannels,”IEEE
Trans.Inf.Theory,vol.55,no.7,pp.3051–3073, Jul.2009.
[2] N. Stolte, “Rekursive Codes mit der Plotkin-Konstruktion und ihre
Decodierung,” Ph.D.dissertation, TUDarmstadt, 2002.
[3] I. Tal and A. Vardy, “List decoding of polar codes,” IEEE Trans. Inf.
Theory,vol.61,no.5,pp.2213–2226, May2015.
[4] E. Arikan, H. Kim, G. Markarian, U¨. Özgu¨r, and E. Poyraz, “Perfor-
mance ofshort polar codes under ML decoding,” in Proc.ICT-Mobile
SummitConf.,Santander, Spain,2009.
[5] P.Trifonov,“Efficientdesignanddecodingofpolarcodes,”IEEETrans.
onCommun.,vol.60,no.11,pp.3221–3227,Nov.2012.
[6] S. B. Korada, “Polar codes for channel and source coding,” Ph.D.
dissertation, Ecole Polytechnique Fédérale de Lausanne, Switzerland,
2009.
[7] J. Guo, A. G.i. Fábregas, and J.Sayir, “Fixed-threshold polar codes,”
inProc.IEEEInt.Symp.Inf.Theory(ISIT),Istanbul,Turkey,Jul.2013,
pp.947–951.