On Puncturing Strategies for Polar Codes

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by  Ludovic Chandesris

On Puncturing Strategies for Polar Codes
Ludovic Chandesris†‡, Valentin Savin†, David Declercq‡
[email protected], [email protected], [email protected]
†CEA-LETI, Minatec campus, Grenoble, France ‡ETIS, ENSEA/UCP/CNRS, Cergy-Pontoise, France
Abstract—This paper introduces a class of specific puncturing optimal solution requires a joint optimization of both P and
patterns, called symmetric puncturing patterns, which can be I, since different puncturing patterns may lead to different
characterized and generated from the rows of the generator
information patterns. Hence, a brute-force search algorithm
matrixG .Theyarefirstshowntobenon-equivalent,thenalow-
N would require testing all the possible puncturing patterns, i.e.
complexitymethodtogeneratesymmetricpuncturingpatternsis
7 pdreopptho,seodv,ewrhtihcehrpoewrfsoromfsGa s.eaSrycmhmtreetericalgpoartittehrmnswairtehfluimrtihteedr o(cid:0)NpNtpi(cid:1)mpalospsaibttielritnieosf, ainnfdorfmorateiaocnhbpiutsn,cet.ugr.i,nbgypuatstienrgntfihneddinegnstihtye
N
1 optimizedbydensityevolution,andshowntoyieldbetterperfor- evolution technique, as explained in Section II. In practice,
0 mance than state-of-the-art rate compatible code constructions,
this is not feasible even for relatively small values of N and
2 relying on either puncturing or shortening techniques.
n Index Terms—Punctured polar codes, equivalent puncturing Np. The optimization problem is slightly different in case of
a patterns, symmetric puncturing patterns. communicationsystemsemployingretransmissiontechniques,
J such as Hybrid Automatic Repeat reQuest (HARQ) [2], [3],
7 I. INTRODUCTION [4], [5], since the overall system performance depends on
1 Polar codes are a recently discovered family of error cor- the performance of both the punctured and the unpunctured
rectingcodes[1],knowntoachievethecapacityofanybinary- (mother)code,whicharefurtherconstrainedbyadependency
]
T input memoryless output-symmetric channel. Their construc- relation,astheybothhavetousethesameinformationpattern
I tion relies on a specific recursive encoding procedure that I.
s. synthesizes a set of N virtual channels from N instances of Whiletheproblemoffindingtheoptimalpuncturingpattern
c the transmission channel, where N denotes the code-length. isstillopen,severalpuncturingtechniqueshavebeenproposed
[
The recursive encoding procedure is reversed at the receiver in the literature, aimed at enhancing the punctured Polar code

1 end,byapplyingaSuccessiveCancellation(SC)decoder.The performance, by either making use of different properties of
v asymptotic effectiveness of the SC decoder derives from the the punctured code (in terms of minimal distance or exponent
6
factthatthesynthesizedchannelstendtobecomeeithernoise- bound)[5],[6],[7],orrelyingonthedensityevolutionanalysis
4
less or completely noisy, as the code-length goes to infinity, [8], [9]. However, as no efficient method has been proposed
7
4 phenomenon which is known as “channel polarization”. to evaluate the optimal solution for moderate to long code-
0 APolarcodewordxN−1 ∈{0,1}N isobtainedbyapplying length, the gap between any method and the optimality is
0
. a linear transformation on a data vector uN−1 ∈ {0,1}N unpredictable.
1 0
0 that contains both information and frozen bits (i.e., bits whose In this paper we introduce a number of theoretical tools
7 valuesareknownatboththetransmitterandthereceiver).The aimed at analyzing and classifying puncturing patterns. To
1 positions of information and frozen bits are specified by a bi- this end, we first introduce equivalent puncturing patterns,
:
v nary information pattern I ⊂{0,1}N, with 1s corresponding and show they yield punctured codes with the same error
i to information bits, and 0s corresponding to frozen bits. The correction performance. Then, we give a constructive solution
X
linear transformation is defined by the square matrix G , of to the problem of finding a unique representative for each
N
r
a size N × N, defined as the n-th Kronecker product of the equivalenceclassofpuncturingpatterns.Sucharepresentative
kernel matrix G [1]: isreferredtoasaprimitivepuncturingpattern,andwefurther
2
(cid:20) (cid:21) distinguishbetweenprimitivepatternsthataresymmetricfrom
1 0
G = (1) those that are not. The symmetry property means that P
2 1 1
is equal to the erasure pattern E that is generated on the
Hence, N =2n and xN−1 =uN−1·G , with G =G⊗n. data vector uN−1, corresponding to virtual channels with
0 0 N N 2 0
However, in practical communication systems the length of symmetriccapacityequaltozero(seeSectionII-B).Then,we
the coded block may not be a power of 2. In this case, show that symmetric patterns can be generated by the rows of
punctured Polar codes have to be considered, in which only a the generator matrix G , and propose an algorithm able to
N
number N − N of the encoded bits xN−1 are transmitted generate symmetric patterns of a given order (the order of a
p 0
over the channel. The positions of the N punctured bits symmetric pattern is defined as the minimum number of rows
p
are indicated by a puncturing pattern P ⊂ {0,1}N. The ofG thatcontributetogeneratingit).Thismakespossibleto
N
optimizationofthepuncturingpatternamountstodetermining implement a brute-force optimization algorithm on symmetric
the N positions yielding the punctured code with the best patternsofpredeterminedmaximumorder.Finally,theperfor-
p
possible decoding performance. It is worth noticing that the mance of the optimized puncturing patterns is evaluated and
compared with state of the art solutions. LetΓ(W[P](i))denotetheprobabilitydistributionfunctionof
N
γ . In practice, Γ(W[P](i)) can be estimated, under the all-
II. PRELIMINARIES i N
zerocodewordassumption,byrelyingonthedensityevolution
A. Notation
(DE) technique [10]. It further allows determining the genie-
• Vectors are denoted by either aN0 −1 = (a0,...,aN−1) aided error probability:
or A = (A(0),...,A(N −1)). Matrices are denoted by
1
boldface uppercase letters, e.g., M. piG|A =Pr(γi <0)+ 2Pr(γi =0), (3)
• B : {0,...,N − 1} → {0,...,N − 1} denotes the
N which can be used as an alternative metric for the choice of
bit-reversal permutation, and B the corresponding binary
N theinformationpatternI,bychoosingbytheK positionswith
permutation matrix, where B (i,j)=1⇔j =B (i).
N N lowest pGA values.
• For any permutation φ:{0,...,N−1}→{0,...,N−1} i|
and any vector A = (A(0),...,A(N −1)), φ(A) is the Definition 1. The data bit erasure pattern is a binary vector
vector defined by φ(A)(i)=A(φ(i)). E[P]∈{0,1}N, such that E[P](i)=1 iff I(W[P](i))=0.
N
• |A| denotes the number of of 1s of a binary vector A.
Assuming that I(W) > 0, it can be shown that E[P]
• A = ∨L(cid:96)=1A(cid:96) denotes the bit-wise OR operation of a set does actually not depend on W, but only on the puncturing
of binary vectors A ,...,A , where A(i) = 1 ⇔ ∃(cid:96) ∈
1 L patternP,asexplainedinSectionIV-A.Whennoconfusionis
{1,...,L}, such that A (i) = 1. By a slight abuse of
(cid:96) possible, we shall simply denote E[P] by E. It follows from
language,weshallalsosaythatAistheunionofthevectors
the above definition that the erasure pattern indicates data bit
A1,...,AL. positionsi∈{0,...,N−1}suchthatΓ(W[P](i))istheDirac
• For two binary vectors A,B, we say that A is included in N
distribution supported at zero. It is clear that the information
B, and write A⊆B, if A∨B =B.
pattern I should be chosen such that E ∧I = 0, where ∧
• A ≤ B denotes the lexicographic ordering of A and B.
lex denotes the bit-wise AND operation, and 0 the all-zero vector.
Thus,assumingA(cid:54)=B,A< B ⇔A(k)<B(k),where
lex The following theorem is proved in [9].
k =min{i|A(i)(cid:54)=B(i)}
Theorem 2. |E[P]|=|P|, for any puncturing pattern P.
B. Punctured Polar Codes
Let uN−1 be a data vector, xN−1 =uN−1·G the corre- C. Successive Cancellation Decoder
0 0 0 N
spondingencodedvector1,andP ⊂{0,1}N apuncturingpat- The SC decoder takes advantage of the code construction,
tern (hence, xi is punctured if and only if P(i)=1). Assume by estimating the value of a data bit ui, denoted by uˆi, based
thatthattransmissiontakesplaceoveraBinary-InputDiscrete onthepreviousestimatesuˆ0i−1.Tothisend,thefollowingLLR
Memoryless Channel (B-DMC) W, with output alphabet Y. value is computed:
We denote by y[P]N0 −1 the punctured received sequence. For (cid:32)Pr(u =0|y[P]N−1,uˆi−1)(cid:33)
stheequseankcee ooffsliemnpgltihcitNy,,wweithshayl[lPa]Nss−um1 e∈th(aYt y∪[P{](cid:63)N0}−)N1 iasnda γˆi =log Pr(uii =1|y[P]00N−1,uˆ0i0−1) , (4)
0
y[P] = (cid:63) ⇔ P(i) = 1 (the output alphabet is extended with 1−sign(γˆ )
i then the data bit estimate is defined by uˆ = i ,
the symbol (cid:63) to account for punctured positions). i 2
LetI ⊂{0,1}N beaninformationpattern,with|I|=K < if I(i) = 1, and uˆi = ui, if I(i) = 0 (note that sign(0) =
N−N .Wedenoteby(N,P,I)thecorrespondingpunctured ±1 with equal probability). Since the SC decoder has to rely
p
Polar code of dimension K and length N −Np. Frozen bits on the previous estimates of the data bits, γˆi and γi values
are assumed to be set to zero, that is u = 0, for I(i) = 0. need not have the same probability distribution. However, in
i
Using similar notation to that of [1], we denote by W[P](i) practice, the Word Error Rate (WER) performance of the SC
N
the virtual channel with input u and output (y[P]N−1,ui−1), decodercanbeapproximatedbythefollowingformula,which
i 0 0
and by I(W[P](i)) the symmetric capacity of W[P](i). For is known to be tight, especially in the high Signal to Noise
N N Ratio (SNR) regime [11]:
a given P, the optimal information set I is given by the K
positions with highest I(W[P](i)) values. WER≈WERGA =1−(cid:89)(1−pGA) (5)
N i|
The symmetric capacity I(W[P](Ni)) can be determined i∈I
from the distribution of the log-likelihood ratio (LLR) values III. EQUIVALENTANDPRIMITIVEPUNCTURINGPATTERNS
γ , computed by the genie-aided2 SC decoder:
i A. Equivalent Puncturing Patterns
(cid:32) (cid:33)
Pr(u =0|y[P]N−1,ui−1) Forj =0,...,n−1andk =0,2j+1,...,(2n−j−1−1)2j+1,
γ =log i 0 0 , (2)
i Pr(ui =1|y[P]N0 −1,ui0−1) lsewtaφpkp,jin:g{t0h,e..e.le,mNe−nt1s}w→ith{in0,t.h.e.,bNlo−ck1s}[kb,e.t.h.e,pke+rm2ujta−tio1n]
1Note that in [1], encoding is defined by xN0−1 = uN0−1·G(cid:101)N, where and [k+2j,...,k+2j+1−1]. Hence:
G(cid:101)N =BN ·GN.Hence,G(cid:101)N differsfromGN byacolumnpermutation,  i, if i<k or i≥k+2j+1
butthetwoconstructionsareknowntobeequivalent. 
2Following[1],werefertosuchadecoderasgenie-aided,sinceitassumes φk,j(i)= i+2j, if k ≤i<k+2j (6)
perfectknowledgeofthepreviousdatabitsui−1.  i−2j, if k+2j ≤i<k+2j+1
0
2
TABLEI Algorithm 1 Function φ
NON-EQUIVALENTPATTERNSWITHSAMEERASURESET
1: procedure φ(P)
punct. I={I(W[P](i))|i=0,...,7} 2: for j =0:1:n−1 do
7
pattern p={pGA|i=0,...,7} [W =BEC(ε=0.5)] 3: for k =0:2j+1 :N −1 do
I={ i0|, 0, 0, 0.25, 0, 0.375, 0.4375,0.9375} 4: if Pkk+2j−1 <lex Pkk++22jj+1−1 then
P p={0.5,0.5,0.5,0.375,0.5,0.3125,0.2813,0.0313} 5: P ←φk,j(P)
I(cid:48)={ 0, 0, 0, 0.25, 0, 0.250, 0.5625,0.9375} 6: end if
P(cid:48) p(cid:48)={0.5,0.5,0.5,0.375,0.5, 0.375, 0.2188,0.0313} 7: end for
P =[11101000], P(cid:48)=[11011000], E[P]=E[P(cid:48)]=P 8: end for
9: return P
10: end procedure
In the following, φ will be referred to as an elementary
k,j
permutation. Clearly, φ ◦φ is the identity permutation.
k,j k,j
is provided in Table I for the Polar code of length N = 8,
Definition 3. Two puncturing patterns P and P(cid:48) are said to
where P = [1,1,1,0,1,0,0,0] and P(cid:48) = [1,1,0,1,1,0,0,0].
be equivalent, denoted by P ≈ P(cid:48), if there exists a sequence
It can be seen that E[P] = E[P(cid:48)] = P, corresponding to
of elementary permutations φ ,...,φ that transform
BN(P) into BN(P(cid:48)). Hence:k1,j1 kt,jt indexes i = 0,1,2,4 with I(W[P]N(i)) = I(W[P(cid:48)](Ni)) = 0.
However, I(W[P](i)) (cid:54)= I(W[P(cid:48)](i)) for i = 5,6, hence P
P(cid:48) =BN(φkt,jt(···(φk1,j1(BN(P)))···)) (7) and P(cid:48) cannot be eNquivalent3. TablNe I also provides the error
probability values pGA. Using Eq. (5), it can be verified that,
Proposition 4. Let P(cid:48) ≈ P, y[P]N−1 ∈ (Y ∪ {(cid:63)})N, i|
0 depending on the number of information bits, better WERGA
and y(cid:48)[P(cid:48)]N−1 be the permuted version of y[P]N−1, defined
0 0 performanceisprovidedbyP(cid:48)ifK =2(withinformationbits
by the permutation B ◦ φ ◦ ··· ◦ φ ◦ B , from
Eq. (7). Let γN−1 andNγ(cid:48)N−k1t,jbte the LLRkv1a,jl1ues coNmputed {u6,u7}),orP ifK =3(withinformationbits{u5,u6,u7}).
0 0 Considering equivalence classes of puncturing patterns is
by the genie-aided SC decoder, under the all-zero codeword
particularly relevant when brute-force search is used to find
assumption, when submitted with y[P]N−1 and y(cid:48)[P(cid:48)]N−1
0 0 the puncturing pattern (together with the corresponding in-
inputs, respectively. Then γN−1 =γ(cid:48)N−1.
0 0 formation set) providing the best WER performance. In the
Proof: It is enough to prove the proposition for t = 1, followingsectionweintroduceaconstructivemethodtodeter-
since B ◦φ ◦···◦φ ◦B =(B ◦φ ◦B )◦ mine a unique representative in each equivalence class, which
··· ◦ (BN ◦φkt,jt ◦B ).kA1,sj1suminNg t = N1, thektp,jrtoposNition allows reducing the complexity of the brute-force search.
N k1,j1 N
is equivalent to the fact that the genie-aided SC decoder on B. Primitive Pattern of an Equivalence Class
G(cid:101)N = BN ·GN is invariant by any elementary permutation For a puncturing pattern P, let φ(P) be the puncturing
φ of y[P]N−1. This can be proved by induction, starting
k,j 0 pattern defined by Algorithm 1. Algorithm 1 visits all the
from the observation that permuting the y and y values
0 1 elementarypermutationsinaspecificorder,butanelementary
on a kernel block (see Fig. 1) does not change the values
permutation φ is performed only if [P(k),...,P(k+2j−
of γ and γ , since under the all-zero codeword assumption k,j
0 1 1)]< [P(k+2j),...,P(k+2j+1−1)]. Note that in such
γ = 2atanh(tanh((cid:96) /2)tanh((cid:96) /2)) and γ = (cid:96) + (cid:96) , lex
0 0 1 1 0 1 a case, φ actually permutes the above two blocks. We
where (cid:96) = log(Pr(x =0|y )/Pr(x =1|y )), i = 0,1. k,j
i i i i i further define:
The induction step follows from the fact that φ permutes
k,j
the half-top and half-bottom outputs of a G(cid:101)2j+1 sub-block of Φ(P)=BN(φ(BN(P)) (8)
G(cid:101)N. (cid:3)
Theorem 6. The following properties hold:
Theorem 5. Let P and P(cid:48) be two equivalent patterns. Then,
(i) Φ(P)≈P
the following properties hold:
(ii) P ≈P(cid:48) ⇔Φ(P)=Φ(P(cid:48))
(i) Γ(W[P](i))=Γ(W[P(cid:48)](i)), ∀W,∀i=0,...,N −1
N N (iii) Φ(P)=argmax B (P(cid:48))
(ii) I(W[P](i))=I(W[P(cid:48)](i)), ∀W,∀i=0,...,N −1 lex N
N N P(cid:48)≈P
(iii) E[P]=E[P(cid:48)]
Fromtheabovetheorem,itfollowsthatΦ(P)onlydepends
Proof: (i) follows from Proposition 4, (ii) follows from (i), ontheequivalenceclassofP.Inthefollowing,wesaythatP
and (iii) from (ii). (cid:3) is a primitive pattern if Φ(P) = P. Hence, each equivalence
In particular, it follows that the genie-aided SC decoder class contains one and only one primitive pattern. The proofs
has the same error probability (pGA,i = 0,...,N −1), and of the above and the following theorem will be provided in
i|
therefore the same WERGA (Eq. (5)), irrespective of which of an extended version of this paper.
P or P(cid:48) is being used.
3In [8], [9], it is stated without proof that E[P] = E[P(cid:48)] if and only if
Itisworthnoticingthatthereexistpuncturingpatternswith pGA=p(cid:48)GA, ∀i=0,...,N−1.Ourexamplealsoshowsthatthestatement
E[P] = E[P(cid:48)], but which are not equivalent. An example ini|[8],[9]i|isactuallyfalse.
3
Fig.1. KernelblocG2,andlistofpuncturinganderasurepatterns
Theorem 7. Let P be a puncturing pattern, P¯ = B (P),
N
P¯ =[P¯(0),...,P¯(N/2−1)]andP¯ =[P¯(N/2),...,P¯(N−
0 1
1)].ThenP isprimitiveifandonlyifB (P¯ )andB (P¯ )
N/2 0 N/2 1
are primitive, and P¯ ≤ P¯ .
1 lex 0
The above theorem states that primitive patterns can be
determined recursively. However, the recursive construction
Fig.2. RecursivestructureofaPolarcodeoflengthN
method requires storing all of the primitive patterns of length
N,inordertoderivethoseoflength2N,thusbecomingunfea-
sibleforlargeN values.InSectionIV,werestricttoaspecific
of E = E[P ] and E = E[P ]. By construction, and
+ + − −
classofpuncturingpatterns,referredtoassymmetric.Weshow
considering the possible erasure patterns for kernel blocks,
that any symmetric pattern is primitive, and propose a search-
it follows that P ⊆ P , and therefore it can be easily
− +
tree algorithm that allows generating symmetric patterns even verified that E ⊆ E . By induction, E = ∨ G(i)
for large values of N. − + − i∈L− N/2
and E =∨ G(i) , with L ⊆L ⊆{0,...,N/2−1}.
+ i∈L+ N/2 − +
IV. SYMMETRICPUNCTURINGPATTERNS It follows that E =∨ G(i), where L=L ∪{N/2+L },
i∈L N + −
A. Symmetric Patterns and also using the fact that G(N/2+i) = (G(i) ,G(i) ), for
N N/2 N/2
Definition8. ApuncturingpatternP issaidtobesymmetric4 any 0≤i≤N/2−1. (cid:3)
if E[P]=P. Note that the proof of the above theorem actually demon-
strates that an erasure pattern is necessarily the the union of a
The following theorem provides a simple characterization of subsetofrowsofG .Italsoindicateshowtheerasurepattern
N
symmetric patterns, as the union of a subset of rows of GN. E[P] can be determined in practice: one has to propagate
Theorem 9. Let P ⊂{0,1}N be a puncturing pattern. Then P, from right to the left, throughout the recursive structure
shown in Fig. 2. At each propagation step, the pattern is
E[P]=P ifandonlyifP =0(theall-zerovector)orP isthe
propagated through a number of kernel blocks, according to
unionofasubsetofrowsofG ,thatis,∃L⊂{0,...,N−1},
N
P = (cid:87) G(i). the P (cid:55)→E[P] rule shown in Fig. 1. In particular, this shows
N that the erasure pattern does not depend on the channel W,
i∈L
but only on the puncturing pattern P.
Proof: First we prove that if P =∨ G(i), then E[P]=P.
i∈L N
Indeed, let L ={0≤i≤N/2−1|i∈L or N/2+i∈L}, Theorem 10. If P is a symmetric pattern, then Φ(P) = P,
+
and L = {0 ≤ i ≤ N/2 − 1 | N/2 + i ∈ L}. Define i.e. P is the primitive pattern of its equivalence class.
−
P = ∨ G(i) and P = ∨ G(i) . Since G(i) ⊂
+ i∈L+ N/2 − i∈L− N/2 N Proof: First, it can be proved that the bit-reversal permutation
G(k), for any k,i, such that G (k,i) = 1, it follows easily of any row of G is also a row of G . Therefore, if P is
N N N N
that P = (P ,P ). Then the equality E[P] = P follows by a symmetric pattern, then so is B (P). Secondly, one can
+ − N
using induction arguments. easilycheckthatanyrowofG ,andthereforeanysymmetric
N
To prove the direct implication, we prove that for any pattern, is invariant by φ. From the above observations, it
P, E[P] is the the union of a subset of rows of G . The follows that for any symmetric pattern P, one has: Φ(P) =
N
proof goes by induction on n. The statement can be easily B (φ(B (P)))=B (B (P))=P. (cid:3)
N N N N
verified for n = 1 (N = 2), according to the list of erasure Duetospacelimitations,theproofofthefollowingtheorem
patterns provided in Fig. 1. Let N = 2n, with n > 1, and will be included in an extended version of the paper.
E = E[P]. Considering the recursive structure from Fig. 2,
Theorem 11. Let P be a puncturing pattern, P¯ = B (P),
the puncturing pattern P generates an erasure pattern on each N
P¯ =[P¯(0),...,P¯(N/2−1)]andP¯ =[P¯(N/2),...,P¯(N−
of the N/2 kernel blocks on the right side of the figure. 0 1
1)]. Then P is symmetric if and only if B (P¯ ) and
Putting them together, they produce two puncturing patterns N/2 0
B (P¯ ) are symmetric, and P¯ ⊆P¯ .
on the top and bottom G blocks, indicated respectively N/2 1 1 0
N/2
by P+ and P−. Hence, E = (E+,E−) is the concatenation While the previous theorem indicates that symmetric pat-
ternscanbedeterminedrecursively(similarlytoprimitivepat-
4NotethatusingArikan’sgeneratormatrixG(cid:101)N =BN ·GN,symmetric terns),thesearch-treealgorithmintroducedinthenextsection
patternscorrespondtopatternsP,suchthatE[P]=BN(P).Suchpatterns is aimed at generating symmetric patterns, by exploiting their
havebeenconsideredin[8],butthelinkwiththerowsofGN (Theorem9)
hasnotbeennotpointedout. descriptionasaunionofasubsetofrowsofGN (Theorem9).
4
TABLEII
B. Search-Tree Algorithm
NUMBEROFPRIMITIVEPATTERNSFORN =64
Theorder λofasymmetricpuncturingpatternP isdefined
as the minimum number of rows of G that generate P. Np 6 8 10 12 14
Hence λ = min(cid:110)|L||P =∪ G(i)(cid:111)N. It can be easily primitive 381 2005 10599 42894 150502
i∈L N symmetric 156 605 2045 5913 14345
verified that a symmetric puncturing pattern P, of order λ
non-symmetric 225 1600 8554 37281 136157
andweight|P|=N ,isgeneratedbyasetLofrowsofG ,
p N
such that:
• |L|=λ and ∀(i,j)∈L2,G(Ni) (cid:54)⊆G(Nj), target word error rate η > 0, we define the noise (variance)
• |∨i∈LG(Ni)|=Np threshold σt2h(P), as follows:
The search-tree algorithm proposed in this section generates σ2(P)=sup(cid:8)σ2 |WERGA(P,σ2)≤η(cid:9) (10)
th
symmetric puncturing patterns of order λ ≤ λ , for some
max
fixed λ value. Note that for small λ values, it might It corresponds to the maximum σ2 for which the punctured
max max
not be possible to get any value of N , thus resulting in a code provides a word error rate less than or equal to η,
p
loss in flexibility with respect to the punctured code-length. assuming an appropriate choice of the information pattern.
However, the flexibility quickly increases with the considered
A. Primitive vs. Symmetric Patterns
maximum order.
For N = 64, we generate all the primitive puncturing
We denote by L = (L(1),...,L(λ )) a vector such
max
patterns, by using the recursive construction method from
that 0 ≤ L(τ) ≤ N, for any τ = 1,...,λ . For
max
Theorem 7. We further class them in two categories, namely
1 ≤ λ ≤ λ , we further denote P = ∨ G(L(τ)).
max λ 1≤τ≤λ N symmetric patterns and non-symetric ones. The number of
Hence, L(1),...,L(λ) are the λ indexes of the G -rows
N patterns in each category is provided in Table II. It can be
contributing to the symmetric puncturing pattern P . Since
λ seen that the number of non-symmetric patterns increases
the rows of G are numbered from 0 to N −1, we define
N much faster than the number of symmetric ones. For all the
G(N) =0, the all-zero vector. We also define P =0.
N 0 puncturing patterns, we use the DE technique5 to determine
For given N and λ values, the search-tree algorithm
p max the noise threshold for a target word error rate η = 10−4,
works as follows:
assuming K = 20 information bits. Fig. 3 shows the noise
0) Let λ=1 and L(τ)=N, for any τ =1,...,λ .
max threshold (in dB) of the best puncturing patterns in each one
1) Calculate the complementary weight values, defined by
of the two categories, for various N values. We observe that
W (i)=|G(i)|−|G(i)∧P |, for i=0,...,N −1. p
cp N N λ−1 the best symmetric and non-symmetric patterns have virtually
2) Determine the largest index i strictly less than L(λ), thesamenoisethresholdforanyN value.Therefore,onemay
p
such that:
restrict the search of good puncturing patterns to symmetric
(cid:26)
0<W (i)≤N −|P |, if λ<λ patternsonly.Moreover,Fig.3alsoshowsthatbyconsidering
cp p λ max (9)
W (i)=N −|P |, if λ=λ only symmetric patterns of order λ ≤ 3, the corresponding
cp p λ max
noise threshold values are very close to the optimal ones.
If no index i verifies Eq. (9), then set λ=λ−1 (upper
This justifies the optimization method considered in the next
tree level) and go to Step 4).
section, which consists in exploring only symmetric patterns
3) Set L(λ)=i;
of relatively low order.
– If |P | = |N |, then store P (symmetric puncturing
λ p λ
pattern of order λ and weight N ), and go back to B. Symmetric vs. SotA Patterns
p
Step 2) (continue the search on the same tree level);
We consider the punctured codes with parameters
– Otherwise, set λ = λ+1, then go back to Step 1)
(N,K,N ) = (256,64,85) and (1024,256,336). For each
p
(continue search on the lower tree level). σ2 value, we rely on DE to find the symmetric puncturing
4) Ifλ=0thenallthepuncturingpatternshavebeenfound
pattern P (and corresponding information pattern I(P,σ2))
and the process stops; otherwise, go to Step 1).
that minimizes WERGA(P,σ2). Due to the very large number
It can be noticed that the complexity of this algorithm is of symmetric patterns, evaluation WERGA(P,σ2) for all of
O(Nλmax), which allows generating symmetric patterns even themisnotfeasible.Thus,werestrictoursearchtosymmetric
for high N values, assuming the λmax is relatively small. patternsoforderλ≤λmax,byusingthesearch-treealgorithm
from Section IV-B. The WER performance of the punctured
V. NUMERICALRESULTS
Polar code under SC decoding is then evaluated by Monte-
Throughout this section, we consider a Binary-Input Ad-
Carlo simulation. Simulation results, are shown in Fig. 4.
ditive While Gaussian Noise (BI-AWGN) channel, with noise
For comparison purposes, we have also included the WER
varianceσ2.ForagivenpuncturingpatternP,theinformation
performance of rate compatible codes (with same (N,N ,K)
p
pattern minimizing the WERGA (Eq. (5)) depends on both
parameters), obtained by using either puncturing [5], [9] or
P and σ2, and can be determined by DE, as explained in
shortening [12] techniques from the state of the art.
SectionII.WedenotethisinformationpatternbyI(P,σ2),and
the corresponding word error rate by WERGA(P,σ2). Given a 5WeuseDEundertheGaussianApproximationhypothesis[11]
5
5
best symmetric 10-1
best non-symmetric
4.8 best symmetric λ ≤ 3
10-2
4.6
N=256
R (dB)th4.4 WER10-3 N=1024
N 4.2
S
10-4
4
Algorithm 4 in [9]
10-5 Shortening [12]
3.8 QUP [5]
Symmetric λ=3
Symmetric λ= {4,5}
3.6 10-6
4 6 8 10 12 14 16 1 1.5 2 2.5 3 3.5 4
Np SNR
Fig.3. Comparisonsofthebestsymmetricandnon-symmetricpatternsfor Fig. 4. WER performance of rate-compatible codes with parameters
N =64,K=20andη=10−4 (N,K,Np)=(256,64,85)and(1024,256,336)
TABLEIII
The number of symmetric patterns of order less than or NUMBEROFSYMMETRICPATTERNSOFORDER≤λmax
equal to λ is given in Table III, for λ = 3,4,5. For
max max λmax 3 4 5
N =256,wenotethatthebestsymmetricpuncturingpatterns
N =256 2940 3.5·105 13.5·106
of order less than or equal to λ , are the same for both
max N =1024 3·105 – –
λ = 4 and λ = 5. Hence, Fig. 4 contains only one
max max
corresponding plot, indicated by λ = {4,5}. This further
max
confirmstheobservationfromtheprevioussection,concerning
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The research leading to these results received funding from CommunicationsLetters,vol.18,no.12,pp.2081–2084,2014.
the European Commission H2020 Programme, under grant
agreement 671650 (mmMagic Project).
6