Polar Codes for Automorphism Ensemble Decoding
PPolar Codes for Automorphism Ensemble Decoding
Charles Pillet, Valerio Bioglio, Ingmar Land
Mathematical and Algorithmic Sciences LabParis Research Center, Huawei Technologies Co. Ltd.Email: { charles.pillet1,valerio.bioglio,ingmar.land } @huawei.com Abstract —In this paper we deal with polar code automor-phisms that are beneficial under low-latency automorphismensemble (AE) decoding, and we propose polar code designs thathave such automorphisms. Successive-cancellation (SC) decodingand thus SC-based AE decoding are invariant with respect tothe only known polar code automorphisms, namely those of thelower-triangular affine (LTA) group. To overcome this problem,we provide methods to determine whether a given polar code hasnon-LTA automorphisms and to identify such automorphisms.Building on this, we design specific polar codes that admitautomorphisms in the upper-diagonal linear (UTL) group, andthus render SC-based AE decoding effective. Demonstrated byexamples, these new polar codes under AE decoding outperformconventional polar codes under SC list decoding in terms of errorrate, while keeping the latency comparable to SC decoding.
Index Terms —Polar codes, code automorphism, successivecancellation, list decoding, permutation decoding, code design.
I. I
NTRODUCTION
Polar codes [1] are a class linear block codes relying on thephenomenon of channel polarization. They are shown to becapacity-achieving on binary memoryless symmetric channelsunder successive cancellation (SC) decoding for infinite blocklength. In the finite-length regime, however, SC decoding suf-fers from poor performance due to error propagation. Variousapproaches have been proposed to overcome this problem. InSC list (SCL) decoding, multiple paths are followed in parallelSC decoders, and the best path is selected from the list atthe end [2]. In connection with an outer CRC, SCL decodingprovides excellent error rate results [3], making this SCL-CRCdecoding the de-facto standard and often used performancebenchmark for polar code decoding.At each bit-decoding step, the SCL decoders need to ex-change information and list operations are required, and thisleads to a large decoding delay. The increase in decoding delaymay be avoided by permutation decoders, where several de-coders run independently in parallel on permuted codewords,and the best candidate is chosen at the end. To this end,permutations based on stage permutations of the polar codeswere exploited: this method is analyzed under SC decoding in[4], while soft cancellation (SCAN) [5] is applied in [6] andbelief propagation (BP) [7] is chosen in [8].Recently, a new type of permutation group decoder forReed-Muller codes [9], [10], termed automorphism ensemble(AE) decoder, has been proposed based on the automorphismgroup of the code [11]. The automorphism group of Reed-Muller codes is known and equivalent to the general affine
Fig. 1. Structure of the automorphism ensemble (AE) decoder. group [12], which allows a large set of automorphisms for AEdecoding. AE decoding may be applied to polar codes giventhe similarity of the two codes; however it requires knowledgeof polar code automorphisms. Lower-triangular affine (LTA)transformations are currently the only known subgroup of theautomorphism group of polar codes [12]. SC decoding wasproved to be invariant under LTA automorphisms, and as aconsequence, SC-based AE decoding exhibits no gain [11].This paper deals with polar-code automorphisms that lieoutside the LTA group. We propose a method to determine if agiven polar code has such automorphisms and to identify suchautomorphism. We further propose a polar code design thataims at constructing polar codes that have sufficiently manynon-LTA automorphisms to allow for efficient AE decoding.Very recently, independently of our research, new automor-phisms of polar codes have been proposed that are generatedby block lower triangular (BLTA) group [13]. As comparedto that contribution, our paper addresses a wider range ofautomorphisms and we specifically focus on upper-diagonallinear (UTL) transformations since they are not invariant withrespect to SC decoding. By error-rate simulations we comparevarious polar codes under SC, SCL and AE decoding. As aresult, newly designed polar codes under AE decoding thatoutperform conventional polar codes under SCL decoding interms of error rate, while the AE decoding latency is only inthe range of SC decoding.II. P
RELIMINARIES
A. Polar Codes A ( N, K ) polar code of length N = 2 n and dimension K isa binary block code defined by the kernel matrix T (cid:44) [ ] , a r X i v : . [ c s . I T ] F e b he transformation matrix T N = T ⊗ n , an information set I ∈ [ N ] and a frozen set F = [ N ] \I , [ N ] = { , , . . . , N − } . Forencoding, an auxiliary input vector u = ( u , u , . . . , u N − ) is generated by assigning u i = 0 for i ∈ F (frozen bits),and storing information in the remaining entries, i ∈ I . Thecodeword is then computed as x = u · T N . The information andfrozen set are typically selected according to the reliabilitiesof the virtual bit-channels resulting from the polarisation.Reliabilities can be determined through different methods [14].Successive Cancellation (SC) decoding is the fundamentaldecoding algorithm for polar codes and is proved to becapacity-achieving at infinite block length [1]. SC list (SCL)decoding was proposed to improve the polar code performancefor finite code lengths [2]. The concatenation of a CRC code[3], used as a genie, provides excellent performance and SCL-CRC is currently the best decoding algorithm of polar codes. B. Monomials codes
Monomial codes of length N = 2 n are a family of codesthat can be obtained as evaluations of boolean functions,namely as polynomials in F [ x , . . . , x n − ] . Polar codes andReed-Muller codes can be described through this formalism[12]. In fact, the rows of T N represent all possible evaluationsof negative monomials over F n [12], where a negative booleanvariable ¯ x i is given by ¯ x i = ¬ x i = (1 ⊕ x i ) . This canbe proven by induction on n = log ( N ) starting from theobservation that the two rows of T are the evaluation oftwo monomials over F , namely the constant monomial ,evaluating to (1 , , and the monomial x , evaluating to (1 , .Table I lists all different negative monomials over F , theirdegrees and their evaluations.A monomial code of length N = 2 n and dimension K is generated by K (positive or negative) monomials outof the N monomials over F n . These K chosen monomialsform the generating monomial set M of the code, whiletheir evaluations, along with the evaluations of their linearcombinations, provide the codebook of the code. A monomialcode is called decreasing if M includes all factors of everymonomial in the set [12]. TABLE IN
EGATIVE MONOMIALS FOR n = 3 AND THEIR EVALUATIONS .Degree Monomial Evaluation Index of row in T N x ¯ x ¯ x x ¯ x ¯ x ¯ x ¯ x ¯ x x ¯ x ¯ x Reed-Muller codes, which may be seen as polar codeswith particular frozen sets, are monomial codes generatedby all monomials up to a certain degree. Polar codes selectgenerating monomials following the polarisation effect; ifthe polar code design is compliant with the universal partial order framework, the resulting code is provably decreasingmonomial [12].
C. Code Automorphisms
An automorphism π of a code C is a permutation thatmaps any codeword x ∈ C into another codeword x (cid:48) ∈ C .The automorphism group Aut ( C ) of a code C is the setcontaining all automorphisms of code C . It is well knownthat the automorphism group of Reed-Muller codes of length N = 2 n is given by the affine transformation group of order n ,GA ( n ) . This group is defined as the set of all transformationsof n variables described by x (cid:55)→ x (cid:48) = Ax + b (1) x, x (cid:48) ∈ F n , where A is an n × n binary invertible matrix and b is a binary column vector of length n .Though the automorphism group of polar codes is un-known, it is proved in [12] that a subgroup of the auto-morphisms of decreasing monomials codes is given by thelower-triangular affine (LTA) group LTA ( n ) . This is the sub-group of GA ( n ) for which A is lower-triangular with oneson the diagonal. Since well-designed polar codes are typicallydecreasing monomials codes, this leads to the knowledge ofone automorphism subgroup of polar codes. D. Automorphism ensemble decoding
In [11], the authors proposed an automorphism ensemble(AE) decoder for Reed-Muller codes, as depicted in Figure 1: M SC-based decoders are run in parallel, each one startingfrom a permuted version of the received codeword where eachpermutation belongs to the automorphism group of the code.The codewords resulting from each SC unit are permuted back,and the most likely candidate is selected.Even though the structure of this decoder is similar to theone of SCL, the main difference is given by the resultingdecoding latency: since the SC-based decoders do not need toexchange information during the decoding process, the latencyof AE is essentially given by the latency of a single SCdecoder. The authors of [11] use BP, SC and SCL algorithmsas inner decoders for AE, resulting in different error rateperformances and latencies.III. A
UTOMORPHISMS OF POLAR CODES
In the following, using the concept of monomial codes, weanalyze and provide new automorphism of polar codes. Asproved in [11], automorphisms within the LTA sub-group areabsorbed by SC decoding. Since permutations in LTA are theonly automorphisms known for polar codes, in order to exploitAE decoders, we need to find new automorphism not in theLTA sub-group. In the following, we will restrict this searchto upper-triangular linear (UTL) transformations, which aretransformations of n variables described by x (cid:55)→ x (cid:48) = U x (2) x, x (cid:48) ∈ F n , where U is an n × n upper-diagonal binary matrixwith full diagonal. This choice is motivated by our observationhat UTL automorphisms are typically not absorbed by SC de-coding and thus useful candidates for SC-based AE decoding. A. Affine transformations as automorphisms
To begin with, we analyze the automorphisms that can beexpressed as affine transformations. Even though it is not clearif affine transformations include all automorphisms of a polarcode (we will provide a counter-example later on), this isa useful starting point to generate UTL automorphisms. Ouranalysis is based on the following Theorem:
Theorem 1.
An affine transformation belongs to the auto-morphism group of a given monomial code if and only if itmaps all generating monomials into linear combinations ofgenerating monomials.
Proof.
This property follows from the definition of auto-morphism of a code. In fact, a permutation belongs to theautomorphism group of a code if and only if it maps everycodeword into another unique codeword, namely if and only ifthe transformed code book is equal to the original one. Code-words of a monomial code correspond to linear combinationsof the generating monomials, while affine transformationscorrespond to permutations.We link this property to the affine transformation matrix A . Non-zero entries of A represent variable changes: setting A i,j = 1 means to substitute variable x i with variable x j .Theorem 2 proves that entry A i,j can be set to only if thevariable change does not impact the generating monomial set. Theorem 2.
A linear transformation x (cid:48) = Ax with matrix A belongs to the automorphism group of the monomial code gen-erated by monomial set M if and only if for every ≤ i, j < n we have that M i,j ⊆ M , where M i,j = M| x i : x i = A i,j · x j represents the set of monomials in M including variable x i where this variable is substituted by A i,j · x j . Proof. If A i,j = 0 , the result of this transformation is theempty set, and hence M i,j ⊆ M . On the other hand, if A i,j = 1 then M i,j represents the monomial set generatedby the monomials in M including variable x i , where thisvariable is substituted by variable x j . If we call M (cid:48) theset of the generating monomials transformed through A , weknow that |M (cid:48) | = |M| , and hence these two sets includethe same number of monomials, since A is a linear bijectivetransformation. Moreover, by construction we have that M (cid:48) iscomposed of linear combinations of elements of M i,j (withthe inclusion of the unity if ∈ M ). As a consequence, thefact that M i,j ⊆ M means that M (cid:48) is composed of linearcombinations of the elements of subsets of M ; this and thefact that |M (cid:48) | = |M| can be true if and only if M (cid:48) = M ,and Theorem 1 concludes the proof.As an example, we consider polar code (16 , generatedby the set of monomials M = { , x , x , x , x x , x x , x } .Entry A , can be put to 1; in fact, if we substitute x by x in M , we have the set of monomials M , = { x , x x , } ⊂M . On the contrary, if we look at A , , we have that TABLE IIV
ARIABLE CHANGE VIA BINARY EXTENSION . indexbinary dec x x x x x x x indexbinary dec x x x x x x x
20 40 60 80 100 120
Code dimension K N u m be r o f ad m i ss i b l e po s i t i on s High-SNR5G sequenceTarget BLER
Fig. 2. Number of upper-triangular admissible positions in matrix A for polarcodes of length N = 128 and dimension K ; three different designs. M , = { x , x x } (cid:54)⊂ M . Similarly, for A , we have M , = { x , x x } (cid:54)⊂ M . Finally, if we substitute anymonomial by x , M i, (cid:54)⊂ M . At the end, the only entryabove the diagonal that may be put to 1 is A , . A subset ofthe automorphism group of the code is then given by the affinetransformation (1) with invertible A and b of the structure A = (cid:63) (cid:63) (cid:63) (cid:63) (cid:63) (cid:63) (cid:63) (cid:63) (cid:63) , b = (cid:63)(cid:63)(cid:63)(cid:63) . (3)Compared to LTA, this increases the automorphisms from1024 to 2688; however, only one of these is a non-trivial UTLtransformation.This procedure can be implemented efficiently using thebinary representation of indices in the information set I . Givena row of the transformation matrix T N , the variables includedin the monomial generating that row are given by the zeroes ofthe binary expansion of the row index, see Table I. The impactof an entry A i,j = 1 can be checked by tracking the changesin the binary expansions of the information bit indices. Allindices in I having x i = 0 in their binary expansions areextracted; then in these indices, x i is set to 1 while x j is setto 0, which represents the change of variables. If the modifiedindices are still in I , then M i,j ⊆ M and A i,j = 1 is feasible.We illustrate this method by continuing the previ-ous example, which has the information set I = { , , , , , , } . For position A , , we have the orig-inal monomials and their extensions in Table II on the left, andthose after variable change x → x on the right. B. Automorphisms of given polar codes
Here we analyse three polar code constructions with respectto their UTL automorphisms, namely the 5G sequence [15],[16], a high-SNR DE/GA design at . dB, and a low-SNRDE/GA design for the target block error rate (BLER) of . . ABLE IIIR
ELATIVE NUMBER OF CODES FOR GIVEN LENGTH N THAT HAVE OR ATLEAST
32 UTL
AUTOMORPHISMS ; TWO DIFFERENT DESIGNS .
128 256 512 1024High-SNR 0 0.0234 0.0508 0.0781 0.1396 ≥ ≥ For a fixed code length N and each code dimension K , wedetermine the number of positions in the upper-triangular (UT)part of matrix A that may be used for automorphisms, termedUT admissible positions, using the methods described above.The results are shown in Fig. 2. The high-SNR design leads toa larger number of admissible positions; the maximal numberof is attained by the eight Reed-Muller codes. For the 5Gsequence, about one quarter of the codes have zero admissibleUT positions. Note that the considered UTL automorphisms,see (2), assume U with full diagonal, and thus t UT admissiblepositions give rise to t UTL automorphisms.An important question is if a code has UTL automorphismsat all and if it has sufficiently many. For the code lengths N = 128 , , , and for each dimension K , wedetermined if the code has UTL automorphisms, whichmakes it not suitable for AE decoding, or at least , whichmakes it suitable. The relative number of codes, for each codelength, fulfilling these criteria is reported in Table III for thehigh-SNR design and for the 5G sequence. The number ofcodes suitable for AE decoding decreases with growing codelength, particularly for the 5G sequence.Not all automorphisms can be expressed as an affine trans-formation. To demonstrate this, we counted the number ofelements of the automorphism group of all polar codes oflength N = 8 by brute force search; the result is reported inTable IV. Here we can see that for most polar codes the num-ber of automorphisms, | Aut | , is larger than the number of affinetransformations, | Aff | . This shows that some automorphismscannot be expressed as affine transformations. As an example,consider the (8 , polar code with I = { , , } , see Table Iand Table IV: swapping 4th and 8th bits is an automorphismbut cannot be expressed as affine transformation. C. UTL design of polar codes
The number of UTL automorphisms for reliability-baseddesigns of polar codes decreases with the code length, asdiscussed above. In the following, we propose a new methodfor polar code design that aims at increasing the number ofUTL automorphisms by modifying the information set of agiven polar code. The method relies on the following corollary:
Corollary 1.
A UTL transformation A belongs to the auto-morphism group of the monomial code generated by monomialset M if and only if M i,j ⊆ M for all ≤ i < j < n − with A i,j = 1 . Proof.
The proof follows from Theorem 2 and from the factthat it is always true that M i,i ⊆ M , i.e., elements on the TABLE IVN
UMBER OF AUTOMORPHISMS OF POLAR CODES OF LENGTH N = 8 . M K | Aut | |
Aff | (0 , , x , x , x , x , x , x (1 , , x , x , x x , x , x , x x , x , x , x , x x , x , x x , x , x , x x , x , x x , x x (2 , diagonal of A need not be checked.For a code of length N = 2 n , assume we have thereliability sequence R , i.e., the sequence of bit-channel indicesin decreasing reliability order. (This is a natural choice, anyother sequence may be used.) For a code of dimension K , thefirst K entries of R are used for the information set I . Toincrease the number of UTL automorphisms, the informationset is modified with the following method.We choose an integer s < K and select as reduced infor-mation set I s the first K − s entries of R ; the correspondinggenerating monomial set is denoted by M s . For a chosentargeted admissible entry A i,j in the upper diagonal part of A , the p ≤ s monomials are determined that are generatedby A i,j = 1 and not in M s yet; this may efficiently be doneusing the methods from the previous section. Adding these p new monomials to M s gives a new polar code of dimension K − s (cid:48) , s (cid:48) = s − p , with the new monomial set M s (cid:48) and the newinformation set I s (cid:48) according to M s (cid:48) . This process is repeateduntil the desired dimension K is reached. The resulting polarcode with information set I has then more UT admissibleentries in A than the original code.As an example, we resume the (16 , polar code presentedin Section III-A, which has one non-trivial element of theUTL (4) subgroup in its automorphism group, see (3). Choos-ing s = 1 , we obtain the (16 , polar code defined by I = { , , , , , } or by M = { , x , x , x , x , x x } .If we call A M s the matrix of dimension n × n having atposition ( i, j ) the monomial(s) needed to be picked to make A i,j part of the automorphism group of the resulting polarcodes, we have: A M = [ ] [ ] [ x x ] [ x x ][ ] [ ] [ x x ] [ x x ][ ] [ ] [ ] [ ][ ] [ ] [ ] [ ] (4) We retrieve the original (16 , polar code by includingmonomial x x , adding the bit-channel index in I . If themonomial x x , representing bit channel , is included in I ,UTL transformations of the form A (6) are automorphisms ofthe new polar code of dimension K = 7 . If monomials x x (bit channel ) or x x (bit channel ) are selected instead,the two (16 , polar have the UTL transformation matrices A (9) and A (5) , allowing either and automorphisms fromUTL (4) , respectively: A (6) = (cid:20) (cid:63) (cid:63) (cid:21) A (9) = (cid:20) (cid:63) (cid:63)
00 1 0 00 0 1 00 0 0 1 (cid:21) A (5) = (cid:20) (cid:63) (cid:63) (cid:63) (cid:21) (5)According to our experience, entries in the lower rightcorner, i.e. U i,j with n/ < i < j , are particularly useful. Eb/N0 [dB] -4 -3 -2 -1 B L E R SCAE32-SC - UTLSC - Optimal SNRAE32-SC - LTASCL32AE4-SCL8 - UTLSCL32 - 5GSCL32-CRC5 - 5G
Fig. 3. Performance of (128 , polar codes designed for high SNR. IV. S
IMULATION R ESULTS
In this section we present simulation results of polar codestransmitted with BPSK modulation over the AWGN channel.As performance reference we use (a) polar codes under SCdecoding, where the code is designed for each SNR value byDE/GA (Optimal SNR), and (b) the 5G polar codes [15], [16]under SCL and SCL-CRC decoding. In the figures, SCL- L refers to SCL decoding with list size L ; AE M -SC and AE M -SCL- L refers to AE decoding with M branches using SC andSCL- L decoding, respectively.First we consider short polar codes designed using DE/GAfor high SNR, i.e. allowing naturally many UTL automor-phisms (Table III); the results are shown in Fig. 3. ForAE decoding, the labels indicate whether LTA or UTL au-tomorphisms are used. As expected from theory [13], LTAautomorphisms provide no gain for AE decoding. The AEdecoders employing UTL automorphisms, however, provideperformance close to SCL decoding. This is a striking result,as the decoding latency is close to that of SC decoding only.Second we consider longer polar codes, for whichreliability-based design leads only to a small number of UTLautomorphisms if to any at all, and we apply our proposedUTL design to ensure sufficiently many UTL automorphisms.The results are shown in Fig. 4. The polar code has length N = 1024 and dimension K = 512 . Note that this dimensionis far from the closest possible Reed-Muller code dimensions,which are and 638, leading to a small number of UTLautomorphisms; for the high-SNR design there are only UTLautomorphisms. The loss inherent to a wrong design SNRdesign grows with the code length; this is confirmed by theloss of . dB of this high-SNR code under SC decoding ascompared to the code with optimal SNR design.We provide two different UTL designs, corresponding totransformation matrices U = (cid:63) (cid:63) (cid:63) U = (cid:63) (cid:63) (cid:63) (cid:63) (cid:63) (6)in order to demonstrate the impact of the positions of theones in the UTL matrix (UT admissible positions) on the Eb/N0 [dB] -5 -4 -3 -2 -1 B L E R SCSC - Optimal SNRSCL32 - 5GSCL32-CRC11 - 5GSC - UTL U SC - UTL U AE8-SC - UTL U AE8-SC - UTL U Fig. 4. Performance of (1024 , polar codes by UTL design. AE decoder. The first UTL design focuses on the upperleft part of the UTL matrix. The polar code is designed byhuman inspection starting from s = 11 , and it has UTLautomorphisms, represented by U . For the second polar codewe aim for ones in the lower right corner of the UTL matrix.Since more bits need to be unfrozen in order to have accessto that area of the UTL matrix, we start with s = 60 . Againdesigned by human inspection, the polar code has UTLautomorphisms, represented by U . The results show thatthe second code has a lower error rate than the first, andwe conjecture that this is due to the UTL automorphismscorresponding to 1s in the lower right corner. Further, thesecond code is better than the 5G polar code under SCLdecoding, though still outperformed by the 5G polar codeunder SCL-CRC decoding.V. C ONCLUSIONS
In this paper we analyze automorphisms of polar codes,specifically automorphisms of the upper-triangular linear(UTL) group. While for short polar codes a conventionalhigh-SNR design using DE/GA already leads to many UTLautomorphisms, longer codes require specific designs. Basedon our automorphism analysis methods, we propose a methodto design polar codes that have sufficiently many UTL auto-morphisms to take advantage of automorphism ensemble (AE)decoding. According to our analysis, the bottom right-part ofthe UTL matrix should be privileged in order to get a polarcode well-designed for AE decoding.The results presented in this paper show that our approachis very promising. Still, many questions remain open for futureresearch. First the proposed design principle is guided byhuman inspection, and an algorithm to fully automatize thecode design is still under research. Second it is not clearwhy the specific positions of the entries in the UTL matrixare so important for AE decoding, and why the bottom rightpart of the UTL matrix has a bigger impact than other parts.Finally, while an additional CRC improves significantly thegain under SCL decoding, no such gain has been noticed underAE decoding, making SCL-CRC decoding still the preferablechoice when no constraints on decoding latency are given. Thereasons of this behaviour are to be understood.
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