Table Of ContentFast Simplified Successive-Cancellation List
Decoding of Polar Codes
Seyyed Ali Hashemi, Carlo Condo, Warren J. Gross
Department of Electrical and Computer Engineering, McGill University, Montre´al, Que´bec, Canada
Email: [email protected],[email protected],[email protected]
Abstract—Polar codes are capacity achieving error correcting devised for SC to SCL, but is based on heuristics and needs
7 codes that can be decoded through the successive-cancellation to be redesigned every time code parameters are modified.
1 algorithm. To improve its error-correction performance, a list- Theauthorsproposedin[8],[9]asphere-basedapproachto
based version called successive-cancellation list (SCL) has been
0 list decoding of polar codes, that has led to the development
proposed in the past, that however substantially increases the
2
numberoftime-stepsinthedecodingprocess.ThesimplifiedSCL and implementation of the simplified successive-cancellation
n (SSCL)decodingalgorithm exploitsconstituent codes withinthe list (SSCL) decoding algorithm [10], [11]. SSCL guarantees
a polar code structure to greatly reduce the required number of significantreductioninthenumberofrequiredtime-stepswith
J time-stepswithoutintroducinganyerror-correctionperformance
respect to SCL without relying on approximations or code-
7 loss. In this paper, we propose a faster decoding approach to
specificdesign.Thus,itcanbeappliedtoanycodeandyields
2 decode one of these constituent codes, the Rate-1 node. We use
thisRate-1nodedecodertodevelopFast-SSCL.Wedemonstrate exactly the same error-correctionperformance of SCL.
] thatonlyalist-size-boundnumberofbitsneedstobeestimatedin Thisworkproposesasimplerdecoderforoneofthespecial
T
Rate-1nodesandFast-SSCLexactlymatchestheerror-correction nodes used in SCL and SSCL, the Rate-1 node. Without any
I performance of SCL and SSCL. This technique can potentially
. kind of error-correctionperformancedegradation,it is able to
s greatly reduce the total number of time-steps needed for polar
c codes decoding: analysis on a set of case studies show that decode a Rate-1 node of length Nν in min(L−1,Nν) time-
[ Fast-SSCLhasanumberoftime-stepsrequirementthatisupto steps,againsttheNν and3Nν−2requiredbySSCLandSCL
66.6% lower than SSCL and 88.1% lower than SCL. respectively.ThisRate-1 nodedecoderis then used instead of
1
v the Rate-1 node decoder of SSCL. We call the decoder that
6 I. INTRODUCTION incorporatesthe newRate-1 decoder“Fast-SSCL”. Giventhat
2 Polar codesare a class of error-correctioncodesintroduced in practical polar codes there are many instances of Rate-1
1 by Arıkan in [1]. They can provablyachievechannelcapacity nodes where L≪N , we show that the proposed Fast-SSCL
ν
8
on a memoryless channel when the code length N tends to can speed up the SSCL decoder of up to 66.6%.
0
. infinity.Thefirstpolarcodedecodingalgorithmtobeproposed The rest of the paper is organized as follows. Section II
1
is the successive-cancellation(SC), that can be representedas briefly introduces polar codes and the SC, SCL and SSCL
0
a binary tree search with complexity O(Nlog N). The full decodingalgorithms.Section III describesthe noveldecoding
7 2
1 search can be completed in 2N −2 time-steps [2]. Various approachforthe Rate-1 node,andprovidesproofof itsexact-
: works in the past [3], [4] have analyzed the nature of the ness. In Section IV the reduction in the number of decoding
v
nodes of the SC tree, noting that nodes whose leaves present time-steps is quantified and compared to previous results for
i
X certain patterns of information and redundancy bits, do not a set of polar codes. Section V draws the conclusions.
r need to be traversed.
a While SC decoding is very effective when applied to polar II. POLARCODES ENCODING AND DECODING
codes with N → ∞, its error-correction performance de- A polar code is represented by P(N,K) and can be
gradesveryquicklywithshortandmediumcodes.Alternative constructed by concatenating two polar codes of length N/2.
decoding algorithms have been proposed to overcome this This recursive construction can be denoted as a matrix mul-
issue, among which successive-cancellation list (SCL) is one tiplication as x = uG⊗n, where u = {u ,u ,...,u }
0 1 N−1
of the most promising [5]: instead of focusing on a single is the sequence of input bits, x = {x ,x ,...,x } is
0 1 N−1
candidate codeword like SC, the L most probable candidate the sequence of coded bits, and G⊗n is the n-th Kronecker
codewords are allowed to survive concurrently. The error- product of the polarizing matrix G = 10 . The encoding
11
correction performance of polar codes under SCL decoding, processinvolvesthe determinationof the K bit-channelswith
(cid:2) (cid:3)
when concatenated with a cyclic redundancy check, has been the best channel characteristics and assigning the information
shown to be comparable to that of some low-density parity- bits to them. The remaining N −K bit-channels are set to a
check codes used in current standards. SCL yields better known value known at the decoder side. They are thus called
error-correctionperformancethan SC at the cost of additional frozen bits with set F. Since the value of these bits does not
latency,requiring2N+K−2 time-stepsto be completed[6], have an impact on the error-correction performance of polar
where K is the number of information bits in the code. The codes on a symmetric channel, they are usually set to 0. The
technique proposed in [7] applies the tree pruning methods codewordxisthenmodulatedandsentthroughthechannel.In
α
β
αl βl αrβr Rep Rate-1
Fig.2. SSCLdecoding treeforP(8,5)and{u0,u1,u2}∈F.
uˆ uˆ uˆ uˆ uˆ uˆ uˆ uˆ thispurpose,apathmetric(PM)isassociatedtoeachpathand
0 1 2 3 4 5 6 7
updated at every new estimation: it can be considered a cost
Fig.1. SCdecodingonabinarytreeforP(8,5)and{u0,u1,u2}∈F. function,and the L pathswith the lowestPMs are allowedto
survive.In the LLR-basedSCL [6], the PM can be computed
as
this paper, we consider BPSK modulation which maps {0,1} i
to {+1,−1}. PMil = ln 1+e−(1−2uˆjl)αjl , (6)
Xj=0 (cid:16) (cid:17)
A. Successive-Cancellation Decoding
where l is the path index and uˆ is the estimate of bit j at
jl
Thebinarytreeshownin Fig.1representsthe SCdecoding path l. A HWF version of Equation (6) has been proposed in
process of P(8,5). For a node of length Nν, soft logarithmic [6]:
likelihood ratio (LLR) values α={α ,α ,...,α } pass
0 1 Nν−1
from parent to child nodes, while the hard bit estimates β = PM−1l =0,
{αβrT0=,hβe{1α,N.2rν.,.αe,rlβe,Nm..νe.−n,1tαs}rvfoelcltoow}rsctahαnel bo=eppc{ooαsmil0t,epαudl1tie,rd.e.ca.tsi,oαnl.N2ν−1} and PMil =(PPMMii−−11ll+, |αil|, iofthuˆeirlw6=ise12,(1−sgn(αil)), (7)
0 1 Nν−1
2 which can be rewritten as
αli =2arctanh tanh α2i tanh αi+2N2ν , (1) PM = 1 i sgn(α )α −(1−2uˆ )α . (8)
(cid:18) (cid:16) (cid:17) (cid:18) (cid:19)(cid:19) il 2 jl jl jl jl
αri =αi+Nν +(1−2βil)αi, (2) Xj=0
2
C. Simplified Successive-Cancellation List Decoding
whereas the N values of β are calculated by means of the
ν
left and right messages βl = {βl,βl,...,βl } and βr = SC decoding requires the traversal of the whole decoding
0 1 Nν−1
{βr,βr,...,βr } as 2 tree. The Fast Simplified SC (Fast-SSC) algorithm in [4]
0 1 N2ν−1 reduces the SC time requirements by exploiting the fact
βl ⊕βr, if i< Nν, that polar codes are constructed by concatenation of smaller
βi = βir i, otherwise2, (3) codes. It identifies different constituent codes which can be
( i−Nν decoded with efficient maximum likelihood decoding tech-
2
where ⊕ is the bitwise XOR operation. Bits in the left and niques, avoiding traversing parts of the decoding tree. In
right child nodes are distinguished by i< Nν. At leaf nodes, particular, Fast-SSC makes use of Rate-0 nodes which have
2
the i-th bit uˆ can be estimated as only frozen bits, Rate-1 nodes consisting of information bits
i
only, Repetition (Rep) nodes which have only frozen bits
uˆ = 0, if i∈F or αi ≥0, (4) except for the rightmost one, and Single Parity-Check (SPC)
i
(1, otherwise. nodes that are made of information bits only except for the
leftmost one. The advantage of Fast-SSC is that not only
Equation(1) can bereformulatedin a morehardware-friendly
it requires fewer time-steps than SC to finish the decoding
(HWF) version that has first been proposed in [2]:
process, but also it provides an exact match to SC with no
αl =sgn(α )sgn(α )min(|α |,|α |). (5) error-correction performance loss.
i i i+Nν i i+Nν
2 2 The SSCL algorithmin [10] providesefficientdecodersfor
B. Successive-Cancellation List Decoding
Rate-0, Rep, and Rate-1 nodes in SCL without traversing the
The error-correction performance of SC when applied to decoding tree while guaranteeing the error-correction perfor-
codes with short to moderate length can be improved by the mance preservation. For example in Fig. 1, the black circles
use of list-based decoding.The SCL algorithmestimates a bit represent Rate-1 nodes, the white circles represent Rate-0
consideringboth its possible values 0 and 1. At every estima- nodes,andthewhitetrianglesrepresentRepnodes.Thepruned
tion, the number of codeword candidates (paths) doubles: in decoding tree of SSCL for the example in Fig. 1 is shown in
orderto limittheincrease inthe complexityofthis algorithm, Fig. 2 which consists of a Rep node and a Rate-1 node.
onlyasetofLcodewordcandidatesismemorizedatalltimes. Let us consider that the vectors α and η = 1−2β are
l l l
Thus,aftereveryestimation,halfofthepathsarediscarded.To relativetothetopofanodeinthedecodingtree.Itwasshown
in[11]thatRate-0nodescanbedecodedinasingletime-step which proves the lemma.
as ThefastRate-1decodercanbesummarizedbythefollowing
Nν−1 theorem and its subsequent proof.
ln 1+e−αil , Exact, (9a)
Theorem 1. In SCL decoding with list size L, the maximum
PMNν−1l=1Xi=N0−1sg(cid:0)n(α )α (cid:1)−α , HWF. (9b) rneuqmubireerdotof gbeitttehsetiemxaatcitosnasmienreasuRltastea-s1thneocdoenvoefnlteionngathlSNCLν
2 il il il decoder is
Rep nodes canbe deXic=o0ded in two time-steps as min(L−1,Nν). (14)
Nν−1 The proposed technique improves the required number of
ln 1+e−ηNν−1lαil , Exact, (10a) time-steps to decode Rate-1 nodes when L−1 < N . Every
ν
PMNν−1l=1Xi=N0ν−1sg(cid:0)n(α )α −η (cid:1) α , HWF. (10b) bointtahfeteLrLthReaLs−1-th can be obtained through hard decision
2 il il Nν−1l il
0, if α ≥0,
where ηNν−1lreprXie=s0ents the bit estimate of the information βil =(1, otheirlwise, (15)
bit in the Rep node. Finally, Rate-1 nodes can be decoded in
without the need for path splitting. On the other hand, in
N time-steps as
ν
case min(L−1,N ) = N , all bits of the node need to be
ν ν
Nν−1 estimatedandthedecodingautomaticallyrevertstotheprocess
ln 1+e−ηilαil , Exact, (11a) describedin[10].Thefollowingproofisneverthelessvalidfor
PMNν−1l=1Xi=N0ν−1s(cid:0)gn(α )α −(cid:1) η α , HWF. (11b) bothPLro−of1:<Nν and L−1≥Nν.
2 il il il il Let us consider the path metrics associated with the
While the SSCL Xai=lg0orithm reduces the number of required {LPMsurv,i.v.in.,gPMpaths }atanbdittheesLtiLmRatvioanluessteapssoiciaatsedPwMitihth=e
time-stepstodecodeRate-1nodesbyalmostafactorofthree, i0 iL−1
Rate-1 node at path l as α = {α ,α ,...,α }. For
it failsto addressthe effectof list size on the maximumnum- l 0l 1l Nν−1l
thepurposeofthisproof,letusalsoconsiderthevectorsPM
ber of required bit estimations. In the following section, we i
and α sorted as follows:
provethatthenumberofrequiredtime-stepstodecodeRate-1 l
nodes depends on the list size and that the new Fast-SSCL PM ≤PM , 0≤l<L−1,
il il+1
algorithmisfasterthanbothSCLandSSCLwithoutincurring
|α |≤|α |, 0≤i<N −1.
any error-correction performance degradation. il i+1l ν
Ateachestimationstepi,thecorrespondingbitisestimated
III. FAST SIMPLIFIED SUCCESSIVE-CANCELLATION LIST
as either 0 or 1, and the PMs are updated as
DECODING
In this section, we propose a fast decoding approach for i
Rate-1 nodes and use it to develop Fast-SSCL. In order to PMil = ln 1+e−ηjlαjl , (16)
prove that it is exact and that no approximationis introduced j=0
X (cid:0) (cid:1)
with respect to SCL and SSCL decoding, we first introduce which is a monotonic and non-decreasing function of i. At
the following lemma. any given estimation step i within the Rate-1 node, the least
Lemma1. Fortwopositiverealnumbersaandbwherea<b, reliable LLR among those still to be estimated is αil.
the following holds: Wenowprovethetheorembycontradiction.Letussuppose
that step L − 1 splits path l into two surviving paths. The
ln 1+e−a +ln 1+eb >ln(1+ea)+ln 1+e−b (12)
corresponding PMs will be
(cid:0)Proof: W(cid:1)e pro(cid:0)ve (cid:1) (cid:0) (cid:1) L−2
ln 1+e−a +ln 1+eb −ln(1+ea)−ln 1+e−b >0. PML−1p= ln 1+e−ηjlαjl +ln 1+e−|αL−1l| , (17)
We(cid:0)can writ(cid:1)e (cid:0) (cid:1) (cid:0) (cid:1) Xj=0 (cid:0) (cid:1) (cid:16) (cid:17)
L−2
ln 1+e−a +ln 1+eb −ln(1+ea)−ln 1+e−b = PML−1q= ln 1+e−ηjlαjl +ln 1+e|αL−1l| , (18)
ln(cid:0) 1+e−a(cid:1) +l(cid:0)n 1+(cid:1)eb = (cid:0) (cid:1) Xj=0 (cid:0) (cid:1) (cid:16) (cid:17)
1+ea 1+e−b where 0 ≤ p < q < L. We now show that there are at least
(cid:18) (cid:19) (cid:18) (cid:19)
1+ea 1+e−b L bit estimation sequences that result in PMs which are less
ln e−a +ln eb = than PM . To this end, we demonstrate that there are L
1+ea 1+e−b L−1q
(cid:18) (cid:19) (cid:18) (cid:19) paths originated from path l with smaller PMs than PM
ln e−a +ln eb =b−a>0, (13) L−1q
that are generated before estimating bit L−1.
(cid:0) (cid:1) (cid:0) (cid:1)
Let us consider the lowest possible value that PM can will lead to PM +|α | ≥ PM and thus to
L−1q i−1l il i−1L−1
assume: a discarded path. Consequently, for all the remaining
estimations in the Rate-1 node, paths need not to be
L−2
PML−1q = ln 1+e−|αjl| +ln 1+e|αL−1l| , (19) duplicated, and bits are estimated as ηil =sgn(αil).
B |α |<PM −PM
Xj=0 (cid:16) (cid:17) (cid:16) (cid:17) il i−1L−1 i−1l
Letusconsiderpositionsp andq in theorderingofPMs
which represents the case where the bits estimated in steps
such that
0≤j ≤L−2matchthe harddecision oftheir corresponding
LLR values, and the L−1-th does not. Let us now consider PM =PM ,
ip i−1l
the bit sequences differing from path q in that the bit that
PM =PM +|α |,
does not match the LLR hard decision is at step w, where iq i−1l il
0≤w ≤L−2,whiletheL−1-thmatches.Thecorresponding where l≤p<q <L. In this case, both bit estimates for
PM would be the least reliable bit have to be taken into account since
theircorrespondingpathswill be orderedamongthe first
L−2
PML−1v = ln 1+e−|αjl| Lto.pInostiutironn,tLheaptastthepinipaonsdititohnusLd−is1caartdsetedp.Ti−he1fiosllmowovinedg
Xj=0 (cid:16) (cid:17)
j6=w estimationstep i+1mustbe evaluatedindependently,to
+ln 1+e|αwl| +ln 1+e−|αL−1l| . (20) see if it falls in case A or B.
Assoon ascase A isencounteredin path l,thatpathdoesnot
(cid:16) (cid:17) (cid:16) (cid:17)
Rewriting Equation (19) as
need to undergo any subsequent splitting, and the remaining
L−2 βil can be obtained through LLR hard decision of Equation
PML−1q = ln 1+e−|αjl| (15). While case B requires continued path splitting, this can
Xj=0 (cid:16) (cid:17) occuralimitedamountoftimesbeforecaseAisencountered.
j6=w The maximumamountof consecutivecase B occurrencescan
+ln 1+e−|αwl| +ln 1+e|αL−1l| , (21) be identified by the following worst case analysis.
(cid:16) (cid:17) (cid:16) (cid:17) 1) Case B occurs at i=0 and l =0.
and using the fact that |α |>|α |, we can use the result
in Lemma 1 to concludeL−1l wl 2) Considering that PM−10 is the PM at l=0 before the
first bit of the Rate-1 node is estimated, if p = 0 and
PM >PM , (22) q =1 then
L−1q L−1v
which in turn results in q > v. Since w can assume L − PM00 =PM−10
1 values, and taking in account the bit sequence represented PM =PM +|α |
01 −10 00
by path p where all the bits agree with their corresponding
PM =PM
LLR hard decision, there are at least L bit sequences which 02 −11
.
resultinasmallerPMthanPML−1q.Therefore,q ≥Lwhich ..
contradicts the assumption that q <L and confirms that path
3) Case B occurs at i=1 and l =0.
q will be discarded. In other words, this proves that paths
4) Since |α |≤|α |, q >1. For p=0 and q =2,
that consider bits not matching the LLR hard decision after 00 10
the L−1-th step will always be discarded: it is thus useless PM =PM
10 −10
to split paths after the L−1-th. Theorem 1 is consequently
PM =PM +|α |
proven. 11 −10 00
PM =PM +|α |
12 −10 10
The proposed theorem remains valid also for the HWF ..
.
formulation that can be written as
5) Since at every step |α | ≤ |α |, then q > i. If at
PM = PMi−1l+|αil|, if ηil 6=sgn(αil), (23) every case B step p=0i−a1n0d q =ii0+1, a total of L−1
il (PMi−1l, otherwise, consecutive case B are possible, after which q >L−1,
At each step i, depending on the value of |α |, two cases resulting in case A. At i=L−2, the L survivingPMs
arise. il after L−1 consecutive case B are the following:
A |αil|≥PMi−1L−1−PMi−1l PML−20 =PM−10
From (23), we can see that the modulusof the least reli-
PM =PM +|α |
ablebit|α |istheminimumquantitythatcanbeaddedto L−21 −10 00
il PM =PM +|α |
tthheanPtMheidnifcfaesreenηciel b6=etswgene(nαtihle).cIufrtrheinstlqyucaonntistyidiesregdrepaattehr .. L−22 −10 10
.
metric PM and the largest surviving path metric
i−1l PM =PM +|α |.
PMi−1L−1, every estimation that sees ηil 6= sgn(αil) L−2L−1 −10 L−20
Much like the case considered in the proof for Theorem 1, 100
the above analysis shows that at most L−1 bit estimations 10−1
are required to guarantee the exact same results as the con-
ventionalSCL. Thus, the theoremis valid also with the HWF 10−2
10−3
Equation (23).
In the presented proof and discussion, α and PM are
l i R R
assumed to be sorted at every step for the sake of simplicity. FE10−4 BE10−5
PMs are sorted every time paths are split, i.e. when an infor-
mation bit is estimated. When the decodingprocess considers
frozen bits, paths are not split and even if modified, PMs 10−6 10−7
retain their ordering. On the other hand, α is not ordered,
l
but at each step i the full vector sorting can be substituted
with the identification of the minimum |αil|. 10−8 10−9
The result of Theorem 1 provides an exact number of 3 4 5 3 4 5
Eb/N0 [dB] Eb/N0 [dB]
bit-estimations in Rate-1 nodes for each list size in SCL
decoding in order to guarantee error-correction performance Fast-SSCL [7]
preservation. The Rate-1 node decoder of [7] states that two
bit-estimations are required to preserve the error-correction Fig.3. FERandBERperformance comparisonofdecoding P(1024,860)
withFast-SSCLandtheempirical methodof[7]whenL=128.Thecyclic
performance, but this result is found empirically. The follow-
redundancy checklengthis32.
ing remarks are the direct results of Theorem 1.
Remark 1. The Rate-1 node decoder of [7] for L = 2 is TABLEI
NUMBEROFTIME-STEPSFORSCL,SSCL,ANDFAST-SSCLDECODING
redundant.
OFAPOLARCODEOFLENGTHN =1024.THECODEISOPTIMIZEDFOR
Theorem1 states thatfora Rate-1nodeoflengthN when Eb/N0=2dB.
ν
L=2, the number of bit-estimations is min(L−1,N )=1.
ν
Therefore, there is no need to estimate the bits after the least Rate L SCL SSCL Fast-SSCL
reliable bit is estimated. [7] for L=2 is thus redundant. 2 2302 533 394
4 2302 533 474
Remark 2. The Rate-1 node decoder of [7] falls short in 1
8 2302 533 518
preserving the error-correction performance for higher rates 4
16 2302 533 532
and larger list sizes. 32 2302 533 533
For codes of higher rates, the number of Rate-1 nodes 2 2558 793 397
of larger length increases [11]. Therefore, when the list size 4 2558 793 500
1
is also large, min(L − 1,Nν) ≫ 2. The gap between the 2 8 2558 793 597
16 2558 793 687
empirical method of [7] and the result of Theorem 1 can
32 2558 793 757
introduce significant error-correctionperformanceloss. Fig. 3
provides the frame error rate (FER) and bit error rate (BER) 2 2814 1001 334
curves obtained with Fast-SSCL decoding (L = 128) for a 3 4 2814 1001 435
8 2814 1001 545
P(1024,860) code. The code is concatenated with a cyclic 4
16 2814 1001 667
redundancy check of length 32, and different curves are
32 2814 1001 801
provided for the Rate-1 node decoder in Theorem 1, and the
empiricalmethodof[7].Itcanbeseenthattheerror-correction
performance loss reaches 0.25dB at FER of 10−5.
TheproposedRate-1nodedecoderisusedintheFast-SSCL be gained through this technique, however, depends on the
algorithm, while the decoders for Rate-0 and Rep nodes structure of the code itself. As an example, Table I shows the
remain similar to those used in SSCL [10]. In the following number of time-steps required to decode a polar code with
section,weshowthatinapolarcode,therearemanyinstances N = 1024 optimized for Eb/N0 = 2 dB: results are given
where L−1<N forRate-1 nodesand using the Fast-SSCL for three different rates, five list sizes, and SCL, SSCL and
ν
algorithm can significantly reduce the number of required Fast-SSCL decoding algorithms.
decoding time-steps with respect to both SCL and SSCL. It can be observed that the required number of time-steps
for SCL and SSCL does not depend on the list size L, but
IV. TIME-STEPREDUCTION just on the code rate, and thus on the number and size of the
InSectionIII,wehavedemonstratedthatwhenL−1<N , constituentcodes.Lowcoderatescanexploitahighernumber
ν
uptoL−1bitestimationsarenecessarywhendecodingRate-1 andlargersize ofRate-0andRep nodes,thusthe reductionin
nodestomatchtheperformanceofSCL andSSCL.Thetime- thenumberoftime-stepsrequiredforSSCLoverSCLreaches
step reduction for the complete polar code decoding that can 76.8% at rate 1/4 against 64.4% at rate 3/4.
Thenumberofdecodingtime-stepsfortheFast-SSCLalgo-
rithm, on the other hand, depends on L: a small list size will
resultinafastdecodingprocess,thatwillhoweveryieldlower
error-correctionperformance with respect to a larger list size,
butwillnotdegradeitwithrespecttoSCLandSSCLwiththe
sameL.Thelargeradvantagescanbeobservedforhighrates,
where Rate-1 nodes are more numerous and are larger: with
L = 2 and rate 3/4, Fast-SSCL requires 66.6% and 88.1%
fewer time-steps than SSCL and SCL respectively, without
causing any deterioration in error-correction performance.
V. CONCLUSION
In this work, we have proposed a faster approach to the
decoding of Rate-1 nodes in polar codes which resulted in
the development of Fast-SSCL decoding algorithm. We have
postulatedanddemonstratedthatthenumberofbitestimations
and consequent path splitting of a Rate-1 node of length N
ν
necessarytoexactlymatchtheerror-correctionperformanceof
SCL or SSCL decoding with list size L is min(L−1,N ).
ν
Consideringasetofcodesasacasestudy,wehaveshownthat
thewholepolarcodedecodingprocesscanbenefitintime-step
reductionofupto88.1%withrespecttoSCL,and66.6%with
respecttoSSCLdecodingalgorithmwithoutanykindoferror-
correction performance degradation.
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