Protograph-Based Design for QC Polar Codes
PProtograph-Based Design for QC Polar Codes
Toshiaki Koike-Akino and Ye Wang
Mitsubishi Electric Research Laboratories (MERL), 201 Broadway, Cambridge, MA 02139, USA.Email: {koike, yewang}@merl.com
Abstract —We propose a new family of polar coding whichrealizes high coding gain, low complexity, and high throughput byintroducing a protograph-based design. The proposed techniquecalled as quasi-cyclic (QC) polar codes can be highly parallelizedwithout sacrificing decoding complexity. We analyze short cyclesin the protograph polar codes and develop a design method toincrease the girth. Our approach can resolve the long-standingunsolved problem that belief propagation (BP) decoding doesnot work well for polar codes due to the inherently short cycles.We demonstrate that a high lifting factor of QC polar codescan improve the performance and that QC polar codes with BPdecoding can outperform conventional polar codes with state-of-the-art list decoding. Moreover, we show that a greedy pruningmethod can improve the performance-complexity trade-off.
I. I
NTRODUCTION
Capacity-approaching forward error correction (FEC) basedon low-density parity-check (LDPC) codes [1]–[14] have madea great contribution to increasing data rates of wireless andoptical communication systems. However, the pursuit of highFEC performance has led to a significant increase in powerconsumption and circuit size. Hence, attaining a good trade-off between performance and computational complexity is ofgreat importance. In addition, recent high-performance LDPCcodes usually require very large codeword lengths, whereasshorter FEC codes are preferred [15] for latency-constrainedsystems, such as Internet-of-Things (IoT) applications.Polar codes [16]–[41] have drawn much attention as alter-native capacity-approaching codes in place of LDPC codesfor short block lengths, in particular for the fifth-generation(5G) networks. Besides encoder design methods [23]–[28], anumber of decoder algorithms were developed [29]–[31]. Withsuccessive cancellation list (SCL) decoding [19], polar codescan be highly competitive with state-of-the-art LDPC codes.To date, various extended versions based on polar coding havealso been proposed in literature; e.g., nonbinary [32], mixed-kernel [33], [34], irregular [35], [36], concatenated [37]–[39],convolutional [40], and turbo product coding [41].In this paper, we introduce a novel family of protograph-based polar codes, which we call quasi-cyclic (QC) polarcodes having circulant permutation at proto-polarization unitsas illustrated in Fig. 1. With a proper circulant shift value, weshow that the QC polar codes can eliminate short cycles in thecode graph, which achieves a remarkable breakthrough towardresolving the long-standing issue that the belief-propagation(BP) decoding does not perform well for polar codes. Inthe QC polar codes, highly parallel short polar codes arecoupled to achieve performance comparable to longer polarcodes while maintaining the computational complexity as low x x x x x x x x x x x x x x x x u u u u u u u u u u u u u u u u F r o z en B i t s I n f o r m a t i on B i t s Proto-Polarization Unit1 st Stage 2 nd rd th Circulant Shift Values
Fig. 1: Four-stage polarization: QC polar codes (2 , , ) having circulant shift values for proto-polarization units.as that of short polar codes. The contributions of this paperare summarized as follows: • Protograph-based polar codes : We propose a new fam-ily of protograph polar codes. To the best of the authors’knowledge, the concept of protograph codes has neverbeen applied to such non-parity-check codes. • QC polar codes : We introduce highly parallelizable QCpolar codes, a special case of protograph, with circulantpermutations at the proto-polarization units. • Complexity analysis : We show that the computationalcomplexity of the proposed QC polar codes can besignificantly decreased with a protograph lifting factor. • Girth analysis : We analyze short cycles of the protographpolar codes, and develop a design method to increase thegirth. Eliminating short cycles enables BP decoding toproperly work for the QC polar codes. • State-of-the-art performance : We demonstrate that theQC polar codes with shallow polarization can achievecompetitive performance of deep polarization codes. • Irregular pruning : Further complexity reduction andperformance improvement are shown with irregular prun-ing of polarization to cut loops in the protograph.II. B
ASICS OF P OLAR C ODES
A. Polar Encoding An n -stage polar code with K information bits and N =2 n encoded bits uses an N × N generator matrix G ⊗ n for a r X i v : . [ c s . I T ] F e b ncoding, where [ · ] ⊗ n denotes the n -fold Kronecker powerand G is a binary kernel matrix defined as G = (cid:20) (cid:21) . (1)Let u = [ u , u , . . . , u N ] T and x = [ x , x , . . . , x N ] T respec-tively denote the column vectors of input bits and encodedbits. The codeword (for non-systematic codes) is given by x = G ⊗ n Bu , where the matrix multiplications are carriedout over the binary field (i.e., modulo- arithmetic), and B denotes an N × N bit-reversal permutation matrix [16]. Dueto the nature of the Kronecker product, polar encoding anddecoding can be performed at a complexity on the orderof O ( N log N ) . The multi-stage operation of the Kroneckerproducts gives rise to the so-called polarization phenomenonto approach capacity in arbitrary channels [16].The polar coding maps the information bits to the K mostreliable locations in u . The remaining N − K input bits arefrozen bits, known to both encoder and decoder. We use K and ¯ K to denote the subsets of { , , . . . , N } that correspondto the information bit and frozen bit locations, respectively.The lowest reliability can be selected to be in ¯ K for frozenbits, e.g., by Bhattacharyya parameter [16], density evolution[23], [24], Gaussian approximation [25], beta expansion [26],genetic algorithm [27], and deep learning [28]. B. Polar Decoding
The original SC decoder [16] proceeds sequentially over thebits, from u to u N . For each index i ∈ { , , . . . , N } , an esti-mate ˆ u i for bit u i is made as follows. If i / ∈ K , then ˆ u i is set tothe known value of u i , otherwise, when i ∈ K , ˆ u i is set to themost likely value for u i given the channel outputs, assumingthat the previous estimates [ˆ u , ˆ u , . . . , ˆ u i − ] are correct. TheSC decoding was improved by the SCL decoder [19], whichproceeds similarly to the SC decoder, except that for each databit index i ∈ K , the decoder retains both possible estimates, ˆ u i = 0 and ˆ u i = 1 , in subsequent decoding paths. The list-decoding approach limits the number of paths to a fixed-sizelist L of the most likely partial paths. The combination of SCLdecoding with an embedded cyclic redundancy check (CRC) toreject invalid paths yields significantly improved performance[19], [37]. Various other decoding algorithms were proposedin the literature, e.g., simplified SC decoding [29], neural SCdecoding [30], and BP list decoding [31]. C. Computational Complexity
It is known that short LDPC codes do not perform wellas shown in [15], where nonbinary (NB) coding can im-prove LDPC codes in the short-length regimes. However, thecomputational complexity of NB-LDPC decoding is generallyhigher than binary counterparts, in particular for large Galoisfield sizes. It is thus of great importance to realize lowcomputational complexity in addition to high coding gain. Weevaluate the computational complexity of polar decoding andshow that it is competitive with LDPC decoding. C o m pu t a t i ona l C o m p l e x i t y pe r C oded B i t Code Block Length N LDPC Code BP ( d v =5)LDPC Code BP ( d v =3)Polar Code SCLQC Polar Code SCL ( Q =256) Typical Irregular LDPC −50%256 fold
Fig. 2: Computational complexity per coded bit as a functionof block length N for standard polar SCL decoding (per list),QC polar BP decoding and LDPC BP decoding (per iteration).The polar SCL decoding has a log-linear complexity; specif-ically, O [ LN log ( N ) / ) for a list size of L . This nonlinearityis a major drawback in comparison to the linear complexityof LDPC BP decoding, i.e., O [2 Id v N ] where d v denotesthe average degree of variable nodes (VNs). Note that thefactor of Id v comes from the bidirectional message passing,whereas SCL decoding uses unidirectional message passingover N log ( N ) / VNs. Due to the nonlinear complexity,polar codes can eventually be less effective than LDPC codesas we increase the block lengths N . However, it turns outto be an advantage when we aim to reduce the block sizes inorder to decrease decoding latency. This is illustrated in Fig. 2,where complexity per coded bit (i.e., divided by N ) is plottedas a function of block length N for polar and LDPC decoding.Because per-bit complexity is constant depending on averagedegree d v for LDPC codes, there is no motivation to decreasethe block length. In contrast, polar decoding becomes simplerwhen we reduce block sizes. Remarkably, polar decoding willbe more efficient than typical LDPC decoding at short blocklengths of N < . This promotes polar codes as a strongcandidate for latency-critical systems.Although the actual computational complexity may vary de-pending on hardware implementation, most prototyping stud-ies [18] have revealed that polar codes can compete favorablywith LDPC codes in terms of complexity. Note that the LDPCdecoding is more complicated for higher rates because theaverage check-node (CN) degree is larger, whereas polar codeshave at most three degrees at CNs. Nevertheless, polar SCLdecoding is not amenable to parallel implementation. In thispaper, we propose a highly parallelizable polar code familywhose complexity is O [ LN log ( N/Q ) / for a parallelismfactor of Q . From Fig. 2, we can observe the significantadvantage in its decoding complexity. The details of ourproposed QC polar codes will be described in the next section. + (b) Q -fold Replication(a) Polar Units (c) Permutation++ ++ ++ Q Parallels ++ ++ ++Circulant Shift: 1Circulant Shift: Q -1 Fig. 3: Lifting operation for proto-polarization units: (a) reg-ular polar units, (b) replication of Q -parallel encoders, (c)permutation for interleaving intermediate encoding bits.III. P ROTOGRAPAH -B ASED
QC P
OLAR C ODES
A. Protograph Codes
Thorpe [6] introduced the concept of protograph codes, aclass of LDPC codes constructed from a protograph in sucha way that the ’s in the parity-check matrix are replacedby ( Q × Q ) -permutation matrices and the ’s by ( Q × Q ) -zero matrices. The permutation size Q is also called a liftingsize. If the permutation matrices are circulant, the protographcode reduces to a well-known QC LDPC code [7]. To thebest of authors’ knowledge, no studies have been reported fordesigning the protographs for polar codes.Analogously in lifting operations of the parity-check matrixfor LDPC codes, we replace the generator matrix of polarcodes. For example, the following generator matrix for -stagepolar codes is replaced with permutation matrices P i,j : G ⊗ = = ⇒ Lifting P , P , P , P , P , P , P , P , P , , where is an all-zero matrix of size Q × Q . The simplestchoice of permutation matrices is a weight- circulant matrix: P i,j = I ( s (cid:48) i,j ) , where I ( s ) denotes the s th circulant permu-tation matrix obtained by cyclically right-shifting a Q × Q identity matrix by s positions, and s (cid:48) i,j is a shift value todesign. For this special case, we may call the protograph polarcodes as QC polar codes. It can be regarded as a generalizedlow-density generator matrix (LDGM) based on polar codes.We consider a hardware-friendly lifting operation at eachpolarization stage with identity diagonal matrices P i,i = I (0) .Our lifting operation is illustrated in Fig. 3, where we replicate Q -parallel polar encoders and permute exclusive-or (XOR)incident bits among the parallel encoders. Fig. 1 shows anexample of QC polar codes having a shift base matrix of size n × n − as follows: S =
139 252 234 156 157 142 50 68134 25 178 20 254 101 146 21279 192 144 129 204 71 237 25237 235 140 72 255 137 203 133 , (2) (a) Cycle-4 Message Path++++ Proto-Check Node s s s s +s -s -s +s (b) Cycle-6 Message Path+ +++ ++++ ++++ +s -s -s -s +s +s Circulant Shift
Fig. 4: Short cycle examples in QC polar codes: (a) cycle- message passing loop, (b) cycle- message passing loop.whose ( i, j ) th shift value is assigned for the j th proto-polarization unit at the i th stage. Note that the QC polarcodes still hold most benefits of original polar codes such asstructured encoding and decoding. As we will discuss below,the QC polar codes have a number of remarkable features. B. High-Girth Design
In order to achieve good performance, we shall design theshift values of QC polar codes. One obviously poor choice isthe case when we use all zeros for shifting, leading to mutuallyindependent Q -parallel short polar codes without any couplinggain. The protograph codes are often designed to achieve ahigh girth — the “girth” of a code is the length of the shortestcycle in the code graph. It is known [11] that the girth of anyconventional QC LDPC code is upper bounded by . Tanner[12] proposed a systematic way to optimize shift values toachieve girth-12. It was further shown in [13] that an irregularQC LDPC code can achieve a girth larger than 12.For QC polar codes (2 n , K, Q ) of code length N = 2 n Q ,there are n n − shift values to design as in (2). Unfortunately,the factor graph of polar codes are inherently loopy and thereexist a large number of short cycles as illustrated in Fig. 4.Nonetheless, by optimizing shift values, we can increase thegirth for QC polar codes with Q > . For example, the cycle- loop in Fig. 4(a) can be eliminated if the shift values satisfythe condition [14]: − s , − s , + s , + s , (cid:54) = 0 (mod Q ) , (3)where we accumulate shift values of all proto-CNs along theloop. Note that the shift values are negated if the path goesdownward. This explains the long-lasting problem that theBP decoding performs very poorly for the conventional polarcodes ( Q = 1 ), i.e., the accumulated shifts will be always zero,resulting in a small girth of . Our QC polar codes resolve thisissue by maximizing the girth in the protograph. Similarly, thecycle- loop in Fig. 4(b) can be removed if we can satisfy − s , − s , + s , + s , + s , − s , (cid:54) = 0 (mod Q ) . Note that irregular polar coding [36] is an alternative that couldremove some, but not all, short-cycle loops. We extended thegreedy design method used for irregular polar coding to jointly −4 −3 −2 −1
0 1 2 3 4 5 6 7 8 90.9dB BE R SNR (dB)Standard Polar Code: q =0QC Polar Code: q =1Optimized QC Polar Code: q =125% Irregular QC Polar Code: q =1 Fig. 5: BER performance of -iteration BP decoding for -stage half-rate QC polar codes (2 , , q ) . Frozen bit indica-tions are [1 , , , .optimize frozen bit locations and circulant shift values bymeans of protograph extrinsic information transfer (P-EXIT)[8], [9] and a hill-climbing girth maximization [14]. C. Error-Rate Performance
Fig. 5 shows bit-error rate (BER) performance as a func-tion of signal-to-noise ratio (SNR) for short polar codes (2 n , n − , q ) with n = 2 polarization stages in additivewhite Gaussian noise channels. We here use -iteration BPdecoding with two-way round-robin scheduling from the firstto the last stages and its reversed direction alternatingly(parallel flooding updates per stage). The first two bits [ u , u ] are frozen. The following shift base matrices are considered: (cid:20) (cid:21) , (cid:20) (cid:21) , (cid:20) (cid:21) , (cid:20) − (cid:21) (4)for standard polar codes ( Q = 2 q = 1 ), un-optimized QCpolar codes, optimized QC polar codes, and irregular QCpolar codes, respectively. We denote a pruned polarization bya negative shift value. As the first two cases do not satisfy thecondition in (3) to eliminate girth- , the BER performanceis worse than the other two cases. By eliminating the cycle- loop, the QC polar codes can achieve a gain of . dBwithout sacrificing any computational complexity. We foundthat frozen bit locations are also important; specifically, nogain was achieved with [ u , u ] being frozen.The performance improvement can be more significant fora deeper stage n and larger lifting size Q . We plot the BERperformance for half-rate -stage QC polar codes (2 , , q ) .We can see that the increase of the lifting size Q = 2 q can significantly improve performance by up to . dB gainover the standard polar codes. It should be noted that theper-bit complexity is identical for all of these QC polarcodes regardless of the lifting size Q ; specifically, Q paralleldecoding of short polar codes requires a total complexity onthe order of Q × O [ n n − ] for a total codeword length of −4 −3 −2 −1
0 1 2 3 4 52.4dB BE R SNR (dB)Standard Polar Code: q =0QC Polar Code: q =1QC Polar Code: q =2QC Polar Code: q =3QC Polar Code: q =8 Fig. 6: BER performance of -iteration BP decoding for -stage half-rate QC polar codes (2 , , q ) . Frozen bit indica-tions are [1 , , , , , , , , , , , , , , , . −4 −3 −2 −1
0 1 2 3 4 BE R SNR (dB)
Standard Polar Code: n =16, q =0 (SCL1)Standard Polar Code: n =16, q =0 (SCL32)QC Polar Code: n =8, q =8 (BP32)QC Polar Code: n =6, q =10 (BP32)Short Polar Code: n =8, q =0 (BP32)Short Polar Code: n =6, q =0 (BP32)QC1024QC256 Fig. 7: BER performance for long half-rate QC polar codes (2 n , n − , q ) of block length N = 2 n + q = 2 . Q × n bits. For Q = 256 , our girth design method couldremove all short cycles up to . The designed shift base matrixis written in (2), and also depicted in Fig. 1.We next demonstrate that our QC polar codes using shallowpolarization stages can compete against long standard polarcodes with deeper polarization stages. Fig. 7 shows the BERperformance of -stage standard polar codes and -/ -stageQC polar codes for a total block length of N = 2 bits.We also present the shallow -/ -stage polar codes withoutprotograph lifting. Noticeably, shallow -stage QC polar codeswith Q = 1024 parallel BP decoding can outperform SCdecoding of -stage polar codes. Furthermore, our -stageQC polar codes with Q = 256 can achieve performancecompetitive with state-of-the-art SCL decoding (with a list sizeof L = 32 ) for the long -stage polar codes. In addition, itwas verified that the QC polar codes can resolve the issuesof BP decoding to offer comparable performance to SCL −4 −3 −2 −1
0 4 8 12 16 20 24 28 32 BE R Number of Pruned Proto−Polarization UnitsStandard Polar Code: n =4, q =0 (3dB)Irregular QC Polar Code: n =4, q =8 (3dB)Standard Polar Code: n =6, q =0 (1dB)Irregular QC Polar Code: n =6, q =10 (1dB) −64%−7% Fig. 8: Impact of pruning proto-polarizations for irregular QCpolar codes (2 n , n − , q ) .decoding with large list sizes. These results are practicallyimpactful because the encoding, decoding, and code design ofshallower polar codes are much simpler and more efficient. D. Irregular Pruning
We finally investigate the irregular QC polar codes whichdeactivate polarization units. As discussed, pruning proto-polarization units may also assist removing short cycles. It wasshown in [36] that the conventional irregular polar codes areoften capable of reducing the encoding/decoding complexity,decoding latency, and even BER (due to improved Hammingweight distributions). Fig. 8 shows performance of QC polarcodes (2 n , n − , q ) when proto-polarization units are grad-ually pruned by a greedy algorithm [36]. For -stage polarcodes, the performance degrades as the number of inactivepolarizations increases. Nevertheless, the QC polar codes with Q = 256 are still better than the standard polar codes ( Q = 1 )until % of the polar units are removed. For -stage polarcodes with Q = 1024 lifting, it was observed that pruningup to % of the proto-polarization units improves the BERperformance over the regular counterpart. In consequence, theirregular QC polar codes can further reduce the decodingcomplexity with potential performance improvement. E. Discussion
Some major advantages of the proposed protograph polarcodes are listed below: • The girth of polar codes can be increased significantly. • The BP decoding can compete with SCL decoding. • Multiple short polar encoders and decoders are imple-mented in a fully parallel fashion with no additionalcomplexity besides circulant message exchanges. • It realizes a low computational complexity equivalent to Q -fold shorter polar codes. • Shallow polarization offers comparable performance todeeper polarization. • Code design is simpler using shallower polarization. • There is a higher flexibility in codeword lengths of nonpowers-of-two, i.e., N = 2 n Q . • Irregular polarization is straightforward to apply with theshift value matrix design. • Well-established techniques such as girth design and P-EXIT from LDPC codes are applicable.We however found that the recent BP list decoding [31]was not compatible with the QC polar codes as it is. Aswe focused on the proof-of-principle study in this paper,there remain many research directions, including extensionsto BP list decoding, systematic encoding, nonbinary codes,systematic circulant shift design, BP scheduling optimization,and multi-weight permutation. In particular, it is interestingto consider inhomogeneous polar codes, e.g., non-identicalfrozen bit locations across Q polar codes, and circulant permu-tations among different polarization stages. We also note thatour QC polar codes are similar to polar product codes [41]in the sense that parallel short polar codes are coupled, butthe fundamental difference lies in its mechanism of coupling(QC polar codes do not need additional row/column polarcodes but computation-free circulant). Also, mixture-kerneland nonbinary polar codes are analogous to the QC polar codesin the sense that a single polarization unit processes multiplebits at once in parallel. However, nonbinary polar codes requireadditional complexity and there is a limited flexibility tochoose a Galois field size Q . More importantly, Q -ary polarcodes have only log ( Q ) bits parallelism, whereas fully Q -parallel encoding is possible in the QC polar codes. We believethat the protograph design for generalized LDGM (includingQC polar codes) will stimulate the research community.IV. C ONCLUSIONS
We proposed a novel class of polar codes called QCpolar codes, which replicate parallel short polar encoders anddecoders with circulant permutations to exchange interme-diate messages among them. We developed a protograph-based design method to optimize the girth to achieve highcoding gain. By removing short cycles, the proposed QCpolar codes can outperform the standard polar codes. Webelieve that the QC polar coding offers a breakthrough toresolve the long-standing issue that BP decoding performspoorly for conventional polar codes due to the inherent girth-4. It was demonstrated that the QC polar codes with shallowpolarization can achieve the state-of-the-art performance ofdeeper polar codes at a considerably reduced complexity. TheQC polar codes are hardware friendly as highly parallel encod-ing/decoding is feasible with a reduced polarization stage. Wealso evaluated the impact of irregular QC polar codes, whichcan further decrease the complexity and BER for some cases.For the proposed protograph codes, we addressed a numberof interesting extensions for future research directions.A
CKNOWLEDGMENT
We would like to thank Drs. David S. Millar, Keisuke Ko-jima, and Kieran Parsons at MERL for their useful discussion.
EFERENCES[1] S. Kudekar, T. Richardson, R. Urbanke, “Threshold saturation via spatialcoupling: Why convolutional LDPC ensembles perform so well over theBEC,”
IEEE Trans. Inform. Theory , vol. 57, pp. 803–34, 2011.[2] B. Smith, M. Ardakani, W. Yu, and F. R. Kschischang, “Design ofirregular LDPC codes with optimized performance-complexity tradeoff,”
IEEE Trans. Commun. , vol. 58, no. 2, pp. 489–499, Feb. 2010.[3] T. Koike-Akino, D. S. Millar, K. Kojima, K. Parsons, Y. Miyata, K.Sugihara, and W. Matsumoto, “Iteration-aware LDPC code design forlow-power optical communications,”
J. Lightw. Technol. , vol. 34, no. 2,pp. 573–581, Jan. 2016.[4] S. ten Brink, G. Kramer, and A. Ashikhmin, “Design of low-densityparity-check codes for modulation and detection,”
IEEE Trans. Com-mun. , vol. 52, no. 4, pp. 670–678, Apr. 2004.[5] M. Ebada, A. Elkelesh, S. Cammerer, and S. ten Brink, “ScatteredEXIT charts for finite length LDPC code design,” arXiv preprint arXiv:1706.09239, June 2017.[6] J. Thorpe, “Low-density parity-check (LDPC) codes constructed fromprotograph,”
IPN Progr. Rep. , pp. 42–154, Aug. 2003.[7] J. Thorpe, K. Andrews, and S. Dolinar, “Methodologies for designingLDPC codes using protographs and circulants,”
IEEE ISIT , p. 236, June2004.[8] G. Liva and M. Chiani, “Protograph LDPC codes design based on EXITanalysis,”
IEEE GLOBECOM , pp. 3250–54, 2007.[9] B.-Y. Chang, L. Dolecek, and D. Divsalar, “EXIT chart analysis anddesign of non-binary protograph-based LDPC codes,”
IEEE MILCOM ,2011.[10] L. Wei, T. Koike-Akino, D. G. Mitchell, T. E. Fuja, and D. J. Costello,“Threshold analysis of non-binary spatially-coupled LDPC codes withwindowed decoding,”
IEEE ISIT , pp. 881–5, 2014.[11] M. Fossorier, “Quasicyclic low-density parity-check codes from circu-lant permutation matrices,”
IEEE Trans. Inform. Theory , vol. 50, no. 8,pp. 1788–93, Aug. 2004.[12] R. M. Tanner, D. Sridhara, and T. E. Fuja, “A class of group-structuredLDPC codes,”
ICIST , July 2001.[13] S. Kim, J. S. No, H. Chung, and D. J. Shin, “Quasi-cyclic low-densityparity-check codes with girth larger than ,” IEEE Trans. Inform. The-ory , vol. 53, no 8, pp. 2885–91, July 2007.[14] Y. Wang, S. C. Draper, and J. S. Yedidia, “Hierarchical and high-girthQC LDPC codes,”
IEEE Trans. Inform. Theory , vol. 59, no. 7, pp. 4553–83, 2013.[15] G. Liva, L. Gaudio, T. Ninacs, and T. Jerkovits, “Code design for shortblocks: A survey,” arXiv preprint arXiv:1610.00873, 2016.[16] E. Arıkan, “Channel polarization: A method for constructing capacity-achieving codes for symmetric binary-input memoryless channels,”
IEEETrans. Inform. Theory , vol. 55, no. 7, pp. 3051–3073, July 2009.[17] E. Arıkan, “Systematic polar coding,”
IEEE Commun. Lett. , vol. 15,no. 8, pp. 860–862, Aug. 2011.[18] G. Sarkis, I. Tal, P. Giard, A. Vardy, C. Thibeault, and W. J. Gross,“Flexible and low-complexity encoding and decoding of systematic polarcodes,”
IEEE Trans. Commun. , vol. 64, no. 7, pp. 2732–2745, July 2016.[19] I. Tal and A. Vardy, “List decoding of polar codes,”
IEEE Trans. In-form. Theory , vol. 61, no. 5, pp. 2213–2226, May 2015.[20] B. Li, H. Shen, and D. Tse, “An adaptive successive cancellationlist decoder for polar codes with cyclic redundancy check,”
IEEECommun. Lett. , vol. 16, no. 12, pp. 2044–2047, Dec. 2012.[21] D. M. Shin, S. C. Lim, and K. Yang, “Mapping selection and codeconstruction for m -ary polar-coded modulation,” IEEE Commun. Lett. ,vol. 16, no. 6, pp. 905–908, June 2012.[22] M. Seidl, A. Schenk, C. Stierstorfer, and J. B. Huber, “Polar-codedmodulation,”
IEEE Trans. Commun. , vol. 61, no. 10, pp. 1302–1306,Oct. 2013.[23] I. Tal and A. Vardy, “How to construct polar codes,”
IEEE Trans. In-form. Theory , vol. 59, no. 10, pp. 6562–6582, Oct. 2013.[24] R. Mori and T. Tanaka, “Performance of polar codes with the con-struction using density evolution,”
IEEE Comm. Lett. , vol. 13, no. 7,pp. 519–21, July 2009.[25] P. Trifonov, “Efficient design and decoding of polar codes,”
IEEETrans. Commun. , vol. 60, no. 11, pp. 3221–27, Nov. 2012.[26] G. He, J. C. Belfiore, I. Land, G. Yang, X. Liu, Y. Chen, R. Li, J.Wang, Y. Ge, R. Zhang, and W. Tong, “Beta-expansion: A theoreticalframework for fast and recursive construction of polar codes,”
IEEEGLOBECOM , Dec. 2017. [27] A. Elkelesh, M. Ebada, S. Cammerer, and S. ten Brink, “Decoder-tailored polar code design using the genetic algorithm,”
IEEETrans. Commun. , vol. 67, no. 7, pp. 4521–34, Apr. 2019.[28] M. Ebada, S. Cammerer, A. Elkelesh, and S. ten Brink, “Deep learning-based polar code design,”
IEEE Allerton , pp. 177–183, Sep. 2019.[29] A. Alamdar-Yazdi and F. R. Kschischang, “A simplified successive-cancellation decoder for polar codes,”
IEEE Commun. Lett. , vol. 15,no. 12, pp. 1378—1380, Oct. 2011.[30] N. Doan, S. A. Hashemi, and W. J. Gross, “Neural successive cancel-lation decoding of polar codes,”
IEEE SPAWC , pp. 1–5, June 2018.[31] A. Elkelesh, M. Ebada, S. Cammerer, and S. ten Brink, “Belief propaga-tion list decoding of polar codes,”
IEEE Commun. Lett. , vol. 22, no. 8,pp. 1536–9, June 2018.[32] R. Mori and T. Tanaka, “Non-binary polar codes using Reed-Solomoncodes and algebraic geometry codes,”
IEEE ITW , 2010.[33] N. Presman, O. Shapira, and S. Litsyn, “Polar codes with mixed kernels,”
IEEE ISIT , pp. 6–10, July 2011.[34] F. Gabry, V. Bioglio, I. Land, and J. Belfiore, “Multi-kernel constructionof polar codes,”
IEEE ICC , 2017.[35] M. El-Khamy, H. Mahadavifar, G. Feygin, J. Lee, and I. Kang, “Relaxedpolar codes,”
IEEE Trans. Inform. Theory , vol. 63, no. 4, pp. 1986–2000,Apr. 2017.[36] T. Koike-Akino, C. Cao, Y. Wang, S. C. Draper, D. S. Millar, K. Kojima,K. Parsons, L. Galdino, D. J. Elson, D. Lavery, and P. Bayvel, “Irregularpolar coding for complexity-constrained lightwave systems,”
J. Lightw.Technol. , vol. 36, no. 11, pp. 2248–58, June 2018.[37] Q. Zhang, A. Liu, X. Pan, and K. Pan, “CRC code design for listdecoding of polar codes,”
IEEE Commun. Lett. , vol. 21, no. 6, pp. 1229–1232, June 2017.[38] Y. Wang, W. Zhang, Y. Liu, L. Wang, and Y. Liang, “An improvedconcatenation scheme of polar codes with Reed–Solomon codes,”
IEEECommun. Lett. , vol. 21, no. 3, pp. 468–71, Dec. 2016.[39] B. Li, H. Shen, and D. Tse, “RM-polar codes,” arXiv preprint arXiv:1407.5483, 2014.[40] B. Bourassa, M. Tremblay, and D. Poulin, “Convolutional polar codes onchannels with memory,” arXiv preprint , arXiv:1805.09378, May 2018.[41] T. Koike-Akino, C. Cao, and Y. Wang, “Turbo product codes withirregular polar coding for high-throughput parallel decoding in wirelessOFDM transmission,”