An Improved SCFlip Decoder for Polar Codes

0.33 MB·

by  Ludovic Chandesris

An Improved SCFlip Decoder for Polar Codes
L.Chandesris†‡, V.Savin†, D.Declercq‡
[email protected], [email protected], [email protected]
†CEA-LETI / Minatec, Grenoble, France ‡ETIS, ENSEA/UCP/CNRS, Cergy-Pontoise, France
Abstract—This paper focuses on the recently introduced Suc- new decoding attempts, in case that a failure of the initial
cessive Cancellation Flip (SCFlip) decoder of polar codes. Our SC decoding attempt is detected by the CRC. Each new
contributionistwofold.First,weproposetheuseofanoptimized
attempt consists in flipping one single decision - starting
metric to determine the flipping positions within the SCFlip
with the least reliable one, according to the absolute value
decoder, which improves its ability to find the first error that
occurred during the initial SC decoding attempt. We also of the log-likelihood ratio (LLR) - of the initial SC attempt,
7 show that the proposed metric allows closely approaching the then decoding the subsequent positions by using standard
1 performance of an ideal SCFlip decoder. Second, we introduce SC decoding. The above procedure is iterated until the CRC
0 a generalisation of the SCFlip decoder to a number of ω nested
is verified or a predetermined maximum number of flips is
2 flips, denoted by SCFlip-ω, using a similar optimized metric to
n determine the positions of the nested flips. We show that the reached. The SCFlip decoder provides an interesting trade-
a SCFlip-2 decoder yields significant gains in terms of decoding off between decoding performance and decoding complexity,
J performance and competes with the performance of the CRC- since each new decoding attempt is only performed if the
7 aided SC-List decoder with list size L = 4, while having an previous one failed. Consequently, the average computational
average decoding complexity similar to that of the standard SC
1 complexity of the SCFlip decoder approaches the one of SC
decoding, at medium to high signal to noise ratio.
decoder at medium to high SNR, while competing with the
Index Terms—Polar Codes, SCFlip decoding, order statistic
] CRC-aided SCL with L = 2, in terms of error correction
T decoding.
performance.
I
. In this work we propose two improvements to the SCFlip
s I. INTRODUCTION
c decoder, aimed at both increasing the error correction perfor-
[ Polar codes are a new class of error-correcting codes, mance and reducing the computational complexity. First, we

1 proposed by Arikan in [1], which provably achieve the ca- propose the use of a new metric to determine the flipping
v pacity of any symmetric binary-input memoryless channel positions within the SCFlip decoder. The proposed metric
6 under successive cancellation (SC) decoding. However, for takesintoaccountthesequentialaspectoftheSCdecoder,and
3
short to moderate blocklengths, the frame error rate (FER) weshowityieldsanimprovedFERperformanceandareduced
7
performance of polar codes under successive cancellation computationalcomplexitycomparedtoLLR-basedmetricused
4
0 decoding does not compete with other families of codes such in [5]. Second, we introduce a generalization of the SCFlip
. as LDPC or Turbo-Codes. decoder to a number of ω nested flips, denoted by SCFlip-
1
0 In [2], a Successive Cancellation list-decoding (SCL) has ω. We show that the SCFlip-2 decoder with the proposed
7 been proposed, which significantly outperforms the simple metric to select the two flipping positions competes with the
1 SC decoding, and approaches the Maximum-Likelihood (ML) CRC-aided SCL decoder with L = 4, in terms of decoding
:
v performance at high signal to noise ratio (SNR). Moreover, performance, while having an average decoding complexity
i whenappliedtopolarcodesconcatenatedwithanoutercyclic similartothatofthestandardSCdecodingatmediumtohigh
X
redundancy check (CRC) code -used to identify the correct SNR. Furthermore, we also usean Oracle-assisted decoder as
r
a message from the decoded list- it has been shown that the in [5] to determine the lower bound of these SCFlip decoders
SCLdecodermaysuccessfullycompetewithotherfamiliesof andshowsthatbothproposedalgorithmsforω =1andω =2
capacity approaching codes, like Low Density Parity Check can closely approach the optimal performance.
(LDPC) codes. However, SCL decoder suffers from high The paper is organised as follows. In Section II, a review
storage and computational complexity, which grows linearly of Polar Codes is presented. Section III describes the SCFlip-
with the size of the list. Several improvements have been ω decoder. Section IV presents the proposed metric for the
proposedtoreduceitscomputationalcomplexity,suchasStack selection of the flipping positions. Simulation results are
decoding (SCS) in [3], but at a cost of an increasing storage presented in section V.
complexity.
II. PRELIMINARIES
Successive Cancellation Flip decoder has been introduced
A. Polar Codes
in [4] for the BEC channel and generalised to binary-input
additive white Gaussian noise (BI-AWGN) channel by using A Polar Code is characterized by the three-tuple (N,K,I),
a CRC in [5]. It is close to the order statistic decoding where N = 2n is the blocklength, K the number of in-
proposed in [6] and which has been specifically used for formation bits and I is the set of K indices indicating the
polar codes in [7]. The idea is to allow a given number of position of the information bits inside the block of size N.
Bits corresponding to positions i(cid:54)∈I are referred to as frozen Algorithm 1 SCFlip decoder with ω =1
bits and are fixed to pre-determined values known at both the 1: procedure SCFLIP-1(Y,I,T(1))
encoder and the decoder. 2: (uˆN0 −1,{L(ui)}i∈I)←SC(Y,I,∅)
We denote U = uN0 −1 the data vector, of length N, 3: if CRC(uˆN0 −1)=success then Return uˆN0 −1 end if
containing K information bits at the positions i ∈ I, and 4: Lflip = FlipDetermine({L(ui)}i∈I,−1,T(1))
N −K frozen bits that are set to zero. The encoded vector, 5: for j =0,..,T(1)−1 do
denoted by X, is obtained by: 6: uˆ0N−1 ←SC(Y,I,Lflip(j))
7: if CRC(uˆN−1)=success then Return uˆN−1 end if
X=U·G 0 0
N 8: end for
whereG isthegeneratormatrixdefinedasin[1].Wefurther 9: Return uˆN−1
N 0
denote by Y the data received from the channel and used at 10: end procedure
the decoder input. Uˆ = uˆN−1 denotes the decoder’s output,
0
with uˆ being the estimation of the bit u .
i i Algorithm 2 SCFlip decoder with ω =2
B. Successive Cancellation Decoder 1: procedure SCFLIP-2(Y,I,T(1),{T(2,1),T(2,2)})
SCdecoderisthestandardlowcomplexitydecoderofpolar 2: (uˆN0 −1,{L(ui)}i∈I)←SC(Y,I,∅)
codes given in [1]. The decoding process consists in taking a 3: if CRC(uˆN0 −1)=success then Return uˆN0 −1 end if
decision on bit ui, denoted uˆi, according to the sign of the 4: L(fl1ip) = FlipDetermine({L(ui)}i∈I,−1,T(1))
LLR: 5: for j =0,..,T(1)−1 do
L(u )=log(cid:18)Pr(ui =0|Y,uˆi0−1)(cid:19) 6: (uˆN0 −1,{L(ui)}I)←SC(Y,I,L(fl1ip)(j))
i Pr(ui =1|Y,uˆi0−1) 7: if CRC(uˆN0 −1)=success then Return uˆN0 −1 end if
8: if j <T(2,1) then
using the decision function h:
9: L(2) =(L(2), FlipDetermine (
(cid:26) flip flip
uˆi =h(L(ui))d=ef 1−signu2(iL(ui)) iiff ii∈∈/ II (1) 10: end if {L(ui)}I,j,L(fl1ip)(j),T(2,2) )
where by convention sign(0) = ±1 with equal probability. 11: end for
Note that the computations and decisions are performed se- 12: for i=0,..,T(2,1)−1 do
quentially in SC decoder, as the estimation of the current bit 13: for j =0,..,T(2,2)−1 do
ui depends on the previous decoded bits uˆi0−1. 14: uˆ0N−1 ←SC(Y{,LI,(1)(i),L(2)(j+i·T(2,2))})
flip flip
III. DEFINITIONANDANALYSISOFSCFLIP-ω DECODERS 15: if CRC(uˆN−1)=success then
0
A. Definition of SCFlip-ω Decoders 16: Return uˆN−1
0
Let C(N,K +r,I) denotes the serial concatenation of an 17: end if
outer (K+r,K) CRC code and an inner (N,K+r,I) Polar 18: end for
code. Note that the number of unfrozen positions of the Polar 19: end for
code is K +r, where K is the number of information bits, 20: Return uˆ0N−1
and r is the size of the CRC. 21: end procedure
TheSCFlipdecoder[5]consistsofastandardSCdecoding,
possibly followed by maximum number T of new decoding
attempts, until no errors are detected by the CRC check. Each thattheglobalstructureoftheSCFlipdecodingstaysthesame,
new decoding consists of (i) flipping only one decision of the the only difference being on the generation of the ordered list
initial SC attempt, then (ii) decoding the subsequent positions of flipping positions, denoted by L . We also introduce a
flip
by using standard SC decoding. The T flipping positions are generalisation of the SCFlip decoder to a number of ω nested
thosecorrespondingtothelowestabsolutevaluesoftheLLRs flips:thefirstflipisperformedononedecisionoftheinitialSC
computed during the initial SC attempt. attempt, while the i-th flip (2≤i≤ω) is performed on a bit
However, the success of the SCFlip decoding depends on position belonging to the new decoding trajectory determined
(i) the ability to find the very first error that occurred during bythepreviousflips(1toi−1).Suchasequenceofω nested
theinitialSCattempt,and(ii)theabilityofSCtosuccessfully flips will also be referred to as an order-ω flip. The SCFlip-ω
decode the subsequent positions, once the first position in decoding for ω = 1 and ω = 2 are presented in Algorithm 1
error has been flipped. In this work we introduce two new and Algorithm 2, respectively.
enhancements to the SCFlip decoder, aimed at improving the AsimpleandefficientimplementationofSCFlip-ω decoder
twoabove-mentionedcharacteristics.Weproposetheuseofan consistsinincrementingωrecursively.Aslongasnoerrorsare
optimized metric to determine the flipping positions, building detected by the CRC code, we proceed step by step from SC
upon the probability of a given position being first error that decoder, SCFlip with 1 flip for a number of T(1) attempts,
occurred in the initial SC attempt (see section IV). We note then SCFlip with 2 flips for a number of T(2) attempts.
2
Algorithm 3 FlipDetermine In [5], an Oracle-assisted decoder (OA-SC) has been in-
1: procedure FLIPDETERMINE({L(ui)}i∈I,k1,T) troduced to count the number ω of channel-generated er-
2: for i=0,...,N −1 do rors which occur during a SC decoding. OA-SC performs
3: if i∈I and i>k1 then m(i)=Mα(ui) a standard SC decoder with a hard decision function h(OA)
4: else m(i)=0 modified to ensure that the decision is correct and no error
5: end if willpropagateduringtheprocess:h(OA)(L(ui))=ui.Hence,
6: end for ω is defined by:
7: J ← sort index(m)
ω = #{h(OA)(L(u ))(cid:54)=h(L(u )},
8: Return J(0:T −1) i i
i∈I
9: end procedure
where the symbol #{A} denotes the number of times the
condition A is verified. In this paper, the parameter ω is
referred to as order of a noise realization.
New decoding attempts in SCFlip decoder are similar to the
Note that we use the same notation ω for the flip order
standard SC decoder, the only difference being on the hard
in the SCFlip-ω decoder and the order of a noise realization
decision function. Thus, we use the notation SC(Y,I,E),
as they are directly related. Indeed, the SCFlip-ω is able to
where E is a set of ω indices corresponding to the flipping
decode a noise realization of order ω(cid:48) ≤ ω, provided that
positions. The hard decision function h(cid:48) can be defined as
(i) the corresponding order-ω(cid:48) flip has been selected in the
follow: corresponding ordered list L(ω(cid:48)) and (ii) the CRC is not
uˆ =h(cid:48)(L(u ))d=ef(cid:26) h(L(ui)) if i∈/ E verified by one of the previousflipdecoding attempts. As in [5],
i i 1−h(L(u )) if i∈E
i weusetheOA-SCdecodertopredicttheoptimalperformance
Note that we have a standard SC decoder if E = ∅. In of a SCFlip-ω decoder, regardless of the choice of the metric
Algorithm 1 and Algorithm 2, candidate positions for the ω- M and the complexity (T(1),T(2),...T(ω)), by declaring a
flips are stored in an ordered list denoted by L(ω) of size decodingfailureifandonlyiftheorderofthenoiserealization
flip
T(ω) and generated by the function FlipDetermine, described is greater than ω. These optimal performance serve as lower
boundsontheFERresultsforpracticalSCFlip-ωdecoders.We
in Algorithm 3, which is performed using a metric denoted
M. The calculation of this metric will be discussed in section will further denote FEROAω the lower bound of the SCFlip-ω
decoder.
IV.
Fig. 1 presents the lower-bounds of SCFlip-ω decoders
Algorithm 3 is used in SCFlip-1 as well as SCFlip-2 to
determineT indexesstrictlylargerthank oftheleastreliable with ω = {1,2,3} over BI-AWGN channel for a Polar code
1
with parameters (N,K+r)=(1024,512+16). We also plot the
decisions according to the proposed metric. To do so, we first
calculatethemetricvectorm,thengenerateanindexvectorJ, performance of the SC decoder with (N,K) = (1024,512).
suchthatm(J)issortedindescendingorder(thisoperationis It can be seen that ideal SCFlip-ω decoders exhibit significant
performed by the function denoted sort index). For SCFlip-1, SNR gains compared to the SC decoder, from 0.5dB for the
L(1) is generated by calling this function once with T =T(1) ideal SCFlip-1, to about 1dB for the ideal SCFlip-2 decoder,
flip at FER=10−4. This ideal performance can be achieved with
and k1 = −1. For the SCFlip-2, we have two degrees of (cid:18)K+r(cid:19)
freedomconcerningthechoiceofthefirstandsecondflipping T(ω) = for any order ω, assuming a perfect CRC,
position. Therefore, we use two parameters T(2,1) and T(2,2) ω
to characterize the flips of order 2. T(2,1) is the number of
positions from the L(1) list, for which flips of order-2 will
flip
be explored. The number of order-2 flips explored for each of
thesepositionsisgivenbyT(2,2).TheorderedlistL(2) ofsize
flip
T(2) is obtained by concatenating the vectors of size T(2,2)
returned by T(2,1) successive calls to this function. For each
call,k correspondstothepositionofthefirstbitflipped.The
1
maximumnumberoforder-2attemptsisT(2) =T(2,1)·T(2,2).
B. Oracle-AssistedDecoderandOrderofaNoiseRealization
Following [5], we distinguish between channel-generated
errors(CGE)andpropagationerrors(PE)intheSCdecoding.
Propagation errors are generated by an erroneous decision,
which propagates in the decoding process, while channel-
generated errors correspond to erroneous decisions which
are only generated by the noise realization at the decoder’s
input. From these definitions, the first error in SC decoding is
necessary a CGE. Fig.1. LowerboundofSCFlip-ω forapolarcode(N,K)=(1024,512)
3
i.e. collision probability equal to 0. In practice, due to non thismetricissub-optimal,becauseitdoesnottakeintoaccount
perfectCRC,theprobabilityofgettingaCRCcollision(hence thesequentialaspectoftheSCdecoder.Indeed,whilealower
an erroneous decoded message) increases with the number LLR absolute value indicates that the corresponding hard
of decoding attempts. Therefore, optimizing the choice of decision has a higher probability of being in error, it does not
flippingpositionsallowsimprovingsimultaneouslythelatency provideanyinformationabouttheprobabilityofbeingthefirst
and the FER performance. error that occurred during the sequential decoding process. To
address this issue, we propose a new metric, which is aimed
C. Importance of SCFlip-1
atidentifyingthefirsterrorthatoccurredduringthesequential
We define PM(ω), the probability of not correcting a noise decoding process. The probability of uˆk being the first error
realization of order ω for a SCFlip-ω using a chosen metric is given by:
M, a number T(ω) of attempts, and assuming a perfect CRC.
k−1
As a consequence, FER of SCFlip-ω can be lower bounded (cid:89)
Pr(uˆ (cid:54)=u ,uˆk−1 =uk−1)=p (uˆ )· (1−p (uˆ ))
by: k k 0 0 err k err i
i=0
(cid:88)ω where perr(uˆi)=Pr(uˆi (cid:54)=ui|Y,uˆi0−1 =ui0−1). This probabil-
FER ≥FER + P (ω(cid:48))·D(ω(cid:48)) (2) ity cannot be computed in practice, as we have no guarantee
SCFlip−ω OAω M
ω(cid:48)=1 that previous bits have been correctly decoded. Instead, we
where D(ω(cid:48)) denote the probability of the noise realization can compute the probability Pr(uˆi (cid:54)= ui|Y,uˆi0−1) given by
being of order ω(cid:48) and FER the lower bound determined (this follows from the definition of L(ui)):
OAω
by the OA-SC defined above. This is an inequality because 1
Pr(uˆ (cid:54)=u |Y,uˆi−1)=
of non perfect CRC. The sum in the right hand side of the i i 0 1+exp(|L(u )|)
i
above inequality is referred to as the loss of order ω. It can
Note that if u is a frozen bit, it cannot be in error, as the
be seen that this loss is incremental, so the loss of order ω i
decoder always takes the right decision. Therefore, for frozen
will propagate at order ω +1. In particular, to construct an
bits, the above probability is set to zero.
SCFlip-2 decoder that closely approaches its theoretical lower
To compute the probability of being the first error, we
bound FER , we would like loss of order 2 to be of same
OA2 consider Pr(uˆ (cid:54)= u |uˆi−1) as an approximation of p (uˆ ),
order of magnitude as FER . This implies that the loss of i i 0 err i
OA2 andintroduceaparameterαtocompensatetheapproximation
order1shouldalsobeofsameorderofmagnitudeasFER ,
OA2 andwhichcanbeoptimizedbysimulation.Thus,theproposed
and therefore order of magnitude smaller than FER . This
OA1 metric is given by:
condition is concretely materialised by a SCFlip-1 decoder
Definition 1: Given a bit u ,k ∈ I, the metric associated
matching its lower bound predicted by the OA-SC decoder. k
to u is defined by:
k
D. Complexity of SCFlip-ω
k−1(cid:18) (cid:19)
1 (cid:89) 1
In addition to the FER performance, the performance of M (u )= ·
α k 1+exp(α|L(u )|) 1+exp(−α|L(u )|)
SCFlip-ω decoder is also characterized by its computational k i=0 i
i∈I
complexity. This computational complexity depends on the (4)
number of decoding attempts performed by the SCFlip-ω where α is a parameter to be optimized by simulation.
decoderinordertodecodeagivennoiserealization.Therefore, We further define the equivalent logarithmic domain metric
we denote by Nc(ave) and call the normalized computational Mα(cid:48)(uk)=−α1 ·log(Mα(uk)). It follows that:
complexity the average number of attempts. It is given by:
k−1
1 (cid:88)
Nc(ave) =1+FERSC·T(ave) SNR−→→+∞1, Mα(cid:48)(uk)=|L(uk)|+ α (1+exp(−α·|L(ui) ) (5)
i=0
i∈I
where T(ave) is the average number of decoding attempts and
=|L(u )|+S (uk)
depends on the metric and the chosen values T(ω). It is worth k α 0
noticing that complexity of SCFlip tends to the one of the SC TheS (uk)sumcanbeseenasapenaltyaddedto|L(u )|,
α 0 k
decoder when SNR tends to infinity. which take into consideration the sequential aspect of the SC
decoding. Indeed, this term increases with increasing number
IV. ANEWMETRICFOREFFICIENTSCFLIP-ω DECODERS
and decreasing reliability of previously decoded bits, so that
In[5],theSCFlipdecoderusesflippingpositionswhichare
thelastdecodedbitsarepenalizedcomparedtothemetric(3).
ordered according to the absolute value of their LLRs. The To understand the impact of the parameter α, we consider the
criterion can be described as a metric M(ui),i ∈ I, defined following limit cases. For α = 0, the above metric becomes
by: M (u ) = 1 , where k is the number of positions in I
M(ui)=|L(ui)| (3) les0s thkan or2keIqual to k.IHence, the M induced ordering
0
The flipping positions are those corresponding to the T(1) corresponds to the usual decoding order. For α −→ +∞
it can be easily seen that lim S (uk) = 0, so that M
positions with the lowest M(ui). However, we point out that α→+∞ α 0 ∞
4
10-1 100
C)
CR10-2 10-1
ct
e
perf 10-2
ng 10-3
mi
su ER10-3
s F
a
1 (10-4
order Metric (3) 10-4 SLoCw deer cbooduenrd SCFlip-1
of Lower bound SCFlip-2
s 10-5 SCFlip-1 Metric (4)
Los Metric (4) Metric (4) 10-5 SSCCFFlliipp--12 MMeettrriicc ((34))
α=0.3 α=0.6 SCL4+CRC8
10-6 10-6
0 10 20 30 40 50 60 70 80 90 100 1 1.5 2 2.5 3 3.5
T(1) SNR
Fig.2. Lossoforder1infunctionofT(1) forSNR=2.5dB Fig.3. PerformanceofSCFlip-1andSCFlip-2decoders
TABLEI
OPTIMIZEDVALUEOFαFORSCFLIP-1ASFUNCTIONOFTHESNRFORA
Thisconditioncaneasilybesatisfiedwiththeproposedmetric
CODE(N,K+r)=(1024,512+16)
(4), while we need T(1) much higher with the metric (3),
SNR(dB) 1.5 2.5 3
whichmeansinpracticehighercomputationalcomplexity.For
αopt 0.4 0.3 0.25
next simulations, we choose T(1) =20.
LetusnowconsiderYbeanoiserealizationoforderω =2
for a given code C(N,K,I) with the first CGE in position
is equivalent to (3). In general, the use of the Mα metric u . We define the set I(cid:48) = {k ∈ I,k > k1} of cardinality
k1
can be seen an intermediate trade-off between the decoding K(cid:48) <K. Consider now the code C (N,K(cid:48),I(cid:48)). The order of
2
order and the one given by the LLRs reliability (3). To find
the noise realization Y is only ω = 1 for the code C and
2
the best possible trade-off, we optimize the value of α by
therefore finding the second CGE is equivalent to decoding a
Monte-Carlo simulation. Note also that the optimized value
noiserealizationoforder1forthecodeC .Asaconsequence,
2
should depend on the code used and the SNR. We provide an
the metric M can be used also for SCFlip-2 by considering
α
intuitive explanation of the behavior of the optimized alpha the set I(cid:48) defined above instead of the set I. However, the
value as function of the SNR. Consider equation (5) for some
optimumvalueforαmaybedifferent.Numericaloptimisation
fixed value of alpha. When the SNR goes to infinity, the
for α for a code (1024,512+16) shows that the optimum
term S (uk) tends to 0 and becomes negligible compared 2
α 0 value is α =0.5>α =0.3 at SNR=2.5dB.
2 1
to |L(u )|, and therefore the sequential characteristic of the
k
decoder is no longer accounted for by the considered metric. V. SIMULATIONRESULTS
Consequently, it is expected that the optimal value of alpha ThroughoutthissectionweconsidertransmissionoveraBI-
will increase with the SNR, so that to rebalance the contribu- AWGN channel, using a CRC-Polar code concatenation with
tion of the Sα(uk0) term to the value of the considered metric. parameters N = 1024, K = 512, and r = 16. The K +r
To confirm this intuition, Table I shows the optimized alpha positions for information and CRC bits, given by the set I,
values for a code (N,K+r)=(1024,512+16) and several are optimized for the SC decoder by Gaussian approximation
SNR values. As expected, it can be observed that the optimal as in [8]. Also, this set is updated for each different value of
alpha value increases with the SNR. the SNR. Concerning the CRC, we use a 16-bit CRC with
We further consider a (1024,528) Polar code and consider generator polynomial g(x)=x16+x15+x2+1.
only random noise realizations of order 1 at SNR= 2.5dB. TheFERperformanceofSCFlip-1andSCFlip-2decodersis
Fig.2plotsthelossoforder1(PM(1)·D(1))assumingperfect showninFig.3.ForSCFlip-1,themaximumnumberofflipsis
CRC. It is calculated as the probability of the only channel- set to T(1) =20. We compare the SCFlip-1 decoder using the
generated error not being in the L(1) list generated by the metric (3), with the one using our proposed metric (4) with
flip
FlipDetermineprocedure,asafunctionofthelistsizeT(1),by α=0.3, and the theoretical lower bound corresponding to an
usingeitherthemetric(3),orourproposedmetric(4)withα= idealSCFlip-1decoder.Whiletheperformancegaincompared
0.3(optimizedvalue)andα=0.6.AccordingtoSectionIII-C, to[5]isnotimpressive,itcanbeseenthatourproposedmetric
wecandeterminethevalueofT(1)suchthatthelossoforder1 closely approaches the theoretical lower bound. As the metric
hasthesameorderofmagnitudeasthetheoreticallowerbound (4) exhibits a negligible loss of order 1, it can further be used
of the SCFlip-2, FER . At SNR=2.5db, the SCFlip-2 lower to correct second order noise realizations (note that SCFlip-1
OA2
boundisabout10−4,therefore,weneedD(1)·P (1)<10−4. and SCFlip-2 using the metric (3) have nearly the same FER
M
5
15
toanidealSCFlipdecoder,andexplainedthatthisanecessary
SCFlip-1 Metric (4)
SSCCFFlliipp--21 MMeettrriicc ((43)) condition for building an effective SCFlip decoder of order 2.
SC
SCL L=2 We have further investigated an SCFlip decoder of order-2,
SCL L=4
which uses an analogous metric to determine the order-2 flip
mplexity10 positions. We have shown that the SCFlip-2 decoder yields
Co significant gains in terms of decoding performance, closely
Normalized acpopdreorawchitihnglistthesizpeerLfo=rm4a,ncwehiolef thhaeviCngRCan-aaidveedragSeCd-Lecisotdidneg-
Average 5 cmoemdpiulemxittoyhsiigmhilSaNr Rto. that of the standard SC decoding at
ACKNOWLEDGMENT
The research leading to these results received funding from
0 the European Commission H2020 Programme, under grant
1 1.5 2 2.5 3 3.5
SNR agreement 671650 (mmMagic Project).
Fig.4. AveragenormalizedcomputationalcomplexityforSCFlip-1 REFERENCES
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achievingcodesforsymmetricbinary-inputmemorylesschannels,”IEEE
performance,sinceinthiscasethelossoforder1isdominant). TransactionsonInformationTheory,vol.55,no.7,pp.3051–3073,2009.
WefurtherplottheFERperformanceoftheproposedSCFlip-2 [2] I.TalandA.Vardy,“Listdecodingofpolarcodes,”InformationTheory,
with T(1) =20,T(2,1) =T(2,2) =5,α =0.3, and α =0.5, IEEETransactionson,vol.61,no.5,pp.2213–2226,2015.
1 2 [3] K.NiuandK.Chen,“Stackdecodingofpolarcodes,”Electronicsletters,
and compare with the CRC-aided SCL with L = 4 and 16- vol.48,no.12,pp.695–697,2012.
bit CRC. The theoretical lower bound of SCFlip-2 decoder is [4] M. Bastani Parizi, “Polar codes: Finite length implementation, error
correlations and multilevel modulation,” Master’s thesis, Swiss Federal
also shown. It can be seen that the proposed SCFlip-2 closely
InstituteofTechnology,2012.
approaches the theoretical lower bound, and exhibits nearly [5] O.Afisiadis,A.Balatsoukas-Stimming,andA.Burg,“Alow-complexity
the same performance as the CRC-aided SCL decoder with improved successive cancellation decoder for polar codes,” in 48th
L=4 at medium/high SNR. AsilomarConferenceonSignals,SystemsandComputers. IEEE,2014,
pp.2116–2120.
In Fig. 4, we plot the average normalized complexity for [6] M.P.FossorierandS.Lin,“Soft-decisiondecodingoflinearblockcodes
SCFlip-1 with metric (3) and T(1)=40, for SCFlip-1 with the based on ordered statistics,” IEEE Transactions on Information Theory,
proposed metric (4) with T(1) =20, and SCFlip-2 with same vol.41,no.5,pp.1379–1396,1995.
[7] D.Wu,Y.Li,X.Guo,andY.Sun,“Orderedstatisticdecodingforshort
parameters as in previous paragraph. Moreover, we also plot polar codes,” IEEE Communications Letters, vol. 20, no. 6, pp. 1064–
thenormalizedcomplexityofSCandSCLdecoderswithL= 1067,2016.
{2,4}. We note that the SCFlip-1 decoder exhibits similar [8] P.Trifonov,“Efficientdesignanddecodingofpolarcodes,”IEEETrans-
actionsonCommunications,vol.60,no.11,pp.3221–3227,2012.
FER performance when using our proposed metric (4) with
T(1) = 20, or the metric (3) with T(1) = 40. We observe
that the complexity of SCFlip is very high at low SNR, but
converges quickly to the one of the SC decoder. For a SNR
of 2.2dB, we have already a complexity lower than the one of
theSCLwithL=2.Moreover,onecanseethattheproposed
metric allows reducing the average normalized complexity of
the SCFlip-1 decoder by a factor of 2, as compared to the
metric (3). Also, computational complexity of our proposed
SCFlip-2 is even slightly better than SCFlip-1 with metric (3)
and T(1) = 40, while it improves performance by 0.4dB at
FER of 10−4.
VI. CONCLUSION
In this paper, we first proposed an improvement of the
SCFlip decoder of order-1, by introducing a new metric to
determine the flipping positions, which takes into account the
sequential aspect of the SC decoder. The proposed metric
increasestheabilityoftheSCFlipdecodertofindthefirsterror
that occurred during the initial SC attempt, thus improving
both decoding performance and computational complexity.
Moreover, we have shown that the proposed metric allows
closelyapproachingthetheoreticallowerboundcorresponding
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