Cluster Pairwise Error Probability and Construction of Parity-Check-Concatenated Polar Codes

Transcription

1 Cluster Pairwise Error Probability and Construction of Parity-Check-Concatenated Polar Codes Tao Wang, Daiming Qu and Tao Jiang, Senior Member, IEEE arxiv: v [cs.it] Oct 08 Abstract A successive cancellation list (SCL) decoder with limited list size for polar codes can not be analyzed as a successive cancellation (SC) decoder, nor as a maximum likelihood (ML) decoder, due to the complicated decoding errors caused by path elimination. To address this issue, an analytical tool, named as cluster pairwise error probability (CPEP), is proposed in this paper to measure the competitiveness of the correct path against the error paths in an SCL decoder. It is shown that the sum of CPEPs over error paths could be used as an indicator of the probability of correct path being eliminated from the decoder list. Then, we use CPEP to explain the error performance gain of parity-check-concatenated (PCC) polar code, and apply CPEP as the optimization criterion in the construction of PCC polar codes, aiming to reduce the elimination probability of the correct path in an SCL decoder with limited list size. Simulation results show that the constructed CRC-PCC polar codes outperform their counterparts of CRC-concatenated polar codes over various codeword lengths, code rates and puncturing patterns. Index Terms Polar codes, successive cancellation list decoding, pairwise error probability (PEP), density evolution, parity check. I. ITRODUCTIO Polar codes [] have been shown to be a family of capacityachieving codes under binary-input discrete memoryless channels (BDMC) with low encoding and decoding complexity, and have recently attracted much attention [] [6]. Although the polar coding theory is impressive, the error performance is not satisfying for the polar codes with short and moderate codeword length under the original successive cancellation (SC) decoding algorithm. To improve the performance, successive cancellation list (SCL) decoder is introduced in []. An SCL decoder maintains L decoding paths in the list, and it approaches the maximum likelihood (ML) decoding performance when L is very large. Based on the SCL decoder, some concatenated polar codes are proposed to further improve the error performance, such as cyclic redundancy check (CRC) concatenated polar codes in [], [3], polar subcodes in [4] and parity-check-concatenated (PCC) polar codes in [5]. The simulation results in [] show that the CRC-concatenated polar codes outperform the low-density parity-check (LDPC) codes used in the WiMAX standard with the same codeword length and code rate. Due to the improved performance, polar codes have recently been adopted as the coding schemes for control channels in 5G wireless communication standards [8]. There exist efficient analytical techniques for error performance evaluation of the SCL decoder with L = (i.e., The authors are with the School of Electronic Information and Communications, Huazhong University of Science and Technology, Wuhan, , China ( qudaiming@hust.edu.cn). conventional SC decoder) and L of large size (in this case, the SCL decoder works nearly as an ML decoder []). For an SC decoder, the density evolution technique is applied to estimate the decision error probabilities over the polarized bit channels [9], from which the upper bound on the frame error rate (FER) of a polar code could be obtained. Afterwards, Tal- Vardy s method [0] and Gaussian approximation [], [] are proposed to fulfill the density evolution with low computation complexity. For an SCL decoder with a large list size, the SCL-based searching algorithms are proposed in [3], [4] to obtain the Hamming weight distribution of a polar code, from which the union bound on FER over AWG channel could be obtained, based on the fact that the SCL decoder with large list size could be treated as an ML decoder. This analytical technique reveals that the key point to improve the error performance of the SCL decoder with the large list size is to increase the minimum Hamming distance of the polar code, and this idea has been reflected in concatenated polar code constructions, such as CRC-concatenated polar codes [3], polar subcodes [4], and PCC polar codes [5], [6]. However, the list size of an SCL decoder might be small or limited in practical applications, due to the consideration on decoding complexity or latency, i.e., a practical SCL decoder would be neither an SC decoder, nor an ML decoder. Therefore, the performance evaluation techniques mentioned above would not be applicable to an SCL decoder with practical list size. Our aim in this paper is to provide an analytical tool for error performance evaluation on polar codes with an SCL decoder of limited list size, since, to the best of our knowledge, this kind of performance evaluation tool is absent. When the list size is limited, the correct path is more likely to be eliminated from the list before the last bit is decided, compared to an ML decoder, and this kind of decoding errors is named as Elimination Error in this paper. Obviously, an elimination error occurs if the number of the error paths with metrics larger than that of the correct path is greater than the list size L, therefore the key to analyze the elimination error is to measure the competitiveness of the correct path against the error paths in the list. The first contribution in this paper is that we propose cluster pairwise error probability (CPEP), which denotes the probability that the path metric of the correct path is smaller than that of a given error path, to measure the competitiveness of the correct path. Furthermore, the method to calculate CPEP is provided, and simulation results show that the sum of CPEPs over error paths could be used as an indicator of the accumulated elimination error probability of an SCL decoder with limited list size. The second contribution in this paper is that we use the

2 CPEP tool to explain why parity bits in a PCC polar code could improve the error performance, and we also propose a suboptimal algorithm to construct the PCC polar code using a criterion based on CPEP. The proposed construction is applied to CRC-PCC polar codes, with various lengths and code rates, where a CRC-PCC polar code is the concatenation of an outer CRC code [] and an inner PCC polar code [5]. The simulation results show that the constructed codes outperform the conventional CRC-concatenated polar codes over various codeword lengths, code rates and puncturing patterns. The remainder of this paper is organized as follows. Section II introduces the SCL decoder and categorizes the SCL decoding errors. Then, we define the CPEP in Section III and provide the CPEP calculation method in Section IV. The PCC polar code construction based on CPEP is introduced in Section V. Section VI presents the simulation results. Finally, we conclude in Section VII. The notations used in this paper is summarized in Table I. II. SCL DECODER AD DECODIG ERRORS The background of polar codes and SCL decoder is introduced in this section. Moreover, we categorize the SCL decoding errors, and introduce the existing performance evaluation techniques for the SCL decoders with different list sizes. A. Polar Codes and SCL Decoder In polar coding [], independent copies of binary-input discrete memoryless channel (BDMC) W : X Y are polarized into bit channels :X Y X i, i =,,...,, where X and Y denote the input alphabet and the output alphabet of the underlying channel W, respectively. It was proved in [] that the sum capacity of bit channels is times that of W, while the capacity of each converges to 0 or with. To achieve high capacity and low error performance, the information bits are assigned to the bit channels with higher capacities. Polar codes are linear block codes and the generator matrix of polar codes with length is G = B F n, where B is a bit-reversal permutation matrix, F = ( 0 ), n = log, and F n denotes n-th Kronecker power of F. The polar codeword c is generated by c = u G, () where c is the shorthand of code bit vector (c, c,, c ) and u = (u, u,, u ) is the encoding bit sequence. The bit indexes of u consist of two parts: one containing the information bits (or unfrozen bits) is denoted by A {,,, }, which corresondes to the bit channels (i =,,, ) of high capacity, the other containing the frozen bits is denoted by A c ={,,, }\A, where \ is used as the set excluding operation in this paper. The frozen bits, generally containing all zeros for symmetric channels, are available at the receiver. In summary, a specific polar code could be identified as a 3-tuple (, M, A), where M is the number of information bits and the dimension of A. The SCL decoder proposed in [] is an improved version of the original SC decoder in []. Instead of keeping only one decoding path as an SC decoder, an SCL decoder with list size L holds the L most likely paths in the list. When decoding the unfrozen bit u i, each path û i,l (l =,,, L) in the list is split into two paths (û i,l, û i,l = 0) and (û i,l, û i,l = ), where û i,l and û i,l denote the decision values of u i and u i, respectively, on the l-th path. Then, the paths with the L largest path metrics are kept in the list and the others are eliminated. The path metric corresponding to the path û i,l is determined as [] (y, û i,l û i,l) = u i+ {0,} i W (y (û i,l, u i+)g ), where y is the received vector and W (y (û i,l, u i+ )G ) is the transition probability of the codeword (û i,l, u i+ )G. When decoding the frozen bit u i, each path û i,l in the list is directly extended to (û i,l, û i,l = 0). After u is decided, the path with the largest path metric is picked out as the decoding result. The error performance of SCL decoding approaches to that of the ML decoder when L is sufficiently large. B. SCL Decoding Errors The decoding errors of an SCL decoder are classified into two categories: Elimination Errors and Picking Errors. Elimination Error is referred to as the decoding error that the correct path is eliminated from the list when deciding an unfrozen bit u i, and there is no correct path in the list after the last bits are decided for all paths. Picking Error is referred to as the decoding errors that the list contains the correct path at the end, however the decoder fails to pick out the correct path as the decoding result. These two types of decoding errors are caused by different reasons. Elimination errors would probably occur especially when the decoder list size is limited or small, where an extreme example is the case when L=. For L=, the SCL decoder is degraded into an SC decoder, and only the elimination errors need to consider. Picking errors would probably occur when the minimum Hamming distance of the polar code is too small. An extreme example that only picking error need to consider is L = M, where the SCL decoder is exactly an ML decoder. In existing studies, the analytical techniques to evaluate the error performance of the SCL decoder are restricted to the cases with L = (i.e., SC decoder) and L = M (or L is large enough, so that the SCL decoder work as an ML decoder). For L =, the elimination errors in an SC decoder are divided into mutually-exclusive events, and the i-th (i =,,, ) event denotes that the first elimination occurs at the bit u i. It is found in [9] that the probability of the i-th event is upper bounded by the decision error probability over the bit channel, which could be estimated via density evolution [9], Tal-Vardy s method [0] or Gaussian approximation [], [] with low complexity. When the decision error probabilities over the bit channels are obtained, the upper bound of FER for an SC decoder is predicted by the sum of the decision error probabilities over the information bits. This ()

3 3 TABLE I OTATIOS otation Dimension Meaning 0 i i the correct path at bit u i in the list when a length- codeword 0 is transmitted. 0 the all-zero codeword of length. A the set containing the indexes of unfrozen bits in a polar code. A c the set containing the indexes of frozen bits in a polar code. B i the event that the correct path is eliminated exactly at the bit u i during SCL decoding. B the bit-reversal permutation matrix for a polar code of length. c a polar codeword of length, i.e., c = (c, c,, c ). C(u i ) a codeword cluster corresponding to ui, i.e., C(ui )={c c =u G,u i+ {0, } i }. F F = ( 0 ). G the generator matrix of polar codes of length. i 0 the index of the first error in the error path ũ i. I the set containing the indexes of M information bits and L CRC CRC bits in u. I k a subset of I and I k =I {,,, p k }. I k (j) the j-th element in I k, where j =,,, I k. Ĩ k a subset of I k. J the number of error paths in the top-j-cpep list. K the number of parity functions in a PCC polar code. L the list size of an SCL decoder. L CRC the CRC length. L (i) (y, 0i, ũi ) L(i) (y,0i, ũi (i) W ) = log (y,ûi =0i u i=0). (y,ûi =ũi u i=ũ i ) M the number of information bits in u. the codeword length. p the punctured codeword length. LE the number of Monte Carlo samples to calculate the mean and variance of [y,(ũj,0),(ũj,)](i.e., LE). CPEP the number of Monte Carlo samples to calculate the CPEP P [C(0 i ) C(ũi ) 0 ]. P the set containing the indexes of K parity bits in u. p k the k-th element in P, where k =,,, K. P [C(ū i ) C(ũi ) c ] the cluster pairwise error probability between the cluster C(ū i ) and C(ũi ), when transmitting the codeword c C(ūi ). R the code rate. S k the search scope of T k. T k the set containing the indexes of information bits corresponding to the k-th parity function. u the encoding bit sequence of a polar code, and u = (u, u,, u ). ũ i i an error path, i.e., ũ i 0i, when 0 is the transmitted codeword in this paper. (y,ûi,l û i,l) the path metric corresponding to the path û i,l in an SCL decoder list. W (y (ûi,l, u i+ )G ) the transition probability of the codeword (û i,l, u i+ )G with a received vector y. y a received vector. σ the noise variance of a BiAWG channel. H \ H H \ H = {i i H, i / H }, where H and H denotes two sets. H the number of elements in H, where H is a set or a vector. g(x) the generator polynomial of a CRC code. E(x) the expected value of x, where x is a random variable representing an LLR in this paper. V (x) the variance of x, where x is a random variable representing an LLR in this paper. max(h) the maximum element in H, where H is a set or a vector. min(h) the minimum element in H, where H is a set or a vector. BitRev(i, ) the bit-reversal map of i, where the length of the binary representation of i is log, for example, BitRev(6, 8) = 3.

4 4 analytical technique for an SC decoder makes contributions to polar code construction, in which the unfrozen bit channel index set A is chosen to minimize the upper bound on FER of an SC decoder. For large list size L (in this case, the SCL decoder works as an ML decoder), the union bound on FER with ML decoding over AWG channel is given as FER d A dq( dsr), where A d denotes the number of codewords with Hamming weight d, SR denotes the signal-to-noise ratio, and Q(x) = + π e u x du. It is found that the performance of the SCL decoder with the large list size is mainly related to Hamming weight distribution of a polar code, which could be obtained efficiently by an SCL-based searching algorithm proposed in [3], [4]. This technique reveals that the key point to improve the performance of polar codes with an SCL decoder of a large list size is to increase the minimum Hamming distance of the polar code, and this idea has been applied in concatenated polar code constructions, such as CRC-concatenated polar codes [3], polar subcodes [4], and PCC polar codes [5], [6]. In practical applications, the list size of an SCL decoder is larger than one, however might be small or limited, due to the consideration of decoding complexity and latency. Thus, the practical SCL decoder would be neither an SC decoder, nor an ML decoder. Therefore, the performance evaluation techniques mentioned above would not be applicable to an SCL decoder with practical list size. The elimination errors in an SCL decoder, similar to those in an SC decoder [9], are divided into mutually-exclusive events B i (i =,,, ), and the event B i represents that the correct path be eliminated exactly at the bit u i during the SCL decoding. From the point of view of path metric, an elimination error at u i indicates that the number of error paths with the metrics larger than that of the correct path is greater than the list size L among the L path candidates generated at u i. Due to the uncertainty of the L error paths at u i, the probability P (B i ) is difficult to be obtained, and to the best of our knowledge, there is no study on the calculation of P (B i ) for an SCL decoder with L >. In our opinion, one key point to understand P (B i ) is to obtain the cluster pairwise error probability (CPEP, see Section III for a precise definition of this quantity), which denotes the probability that the path metric of the correct path is smaller than that of an given error path when deciding a particular bit (u i ). Apparently, the sum of CPEPs over error paths reflects the competitiveness of the correct path at the bit u i, i.e., the smaller the sum of CPEPs, the more competitive the correct path at u i, and the smaller the accumulated elimination error probability P (B i ) could be. The proposed CPEP tool makes one step forward towards the understanding of P (B i ) and the elimination error analysis of SCL decoders. In addition, we apply the CPEP tool in the PCC polar code construction, to demonstrate its practical value. III. CLUSTER PAIRWISE ERROR PROBABILITY In this section, the cluster pairwise error probability (CPEP) is proposed to measure the competitiveness of the correct path against a given error path. Before we define the CPEP, we firstly explain the word cluster. y y c : the codeword in the same cluster as c (a) codeword pairwise error (b) cluster pairwise error c c dc (, c ) c C( u i ) C( u i ) c : transmitted codeword c : distorted codeword y : received vector c c Fig.. Pairwise errors of two codewords and two clusters. Definition (Codeword Cluster): Given a polar code (, M, A), a polar codeword cluster of u i (i =,,, ) is defined as a codeword set C(u i )={c c =u G, u i+ {0, } i }. According to Definition, the codeword cluster C(u i ) is only identified by a bit sequence u i, which also denotes a decoding path at Bit i in an SCL decoder list. Therefore, each path in a decoder list is corresponding to a codeword cluster. Definition (Cluster Pairwise Error Probability, CPEP): Given a polar code (, M, A), two different codeword clusters C(ū i ) and C(ũ i )(ũ i ū i ), and a transmitted polar codeword c C(ū i ), the pairwise error probability between C(ū i ) and C(ũ i ) is defined as P [C(ū i ) C(ũ i ) c ] = P W (y c ) < c C(ūi ) W (y c ), c C(ũi ) where c and c denote the elements in C(ū i ) and C(ũ i ), respectively, y denotes the received vector of the transmitted codeword c, W (y c ) and W (y c ) denote the transition probabilities of c and c, respectively. The concept of CPEP, in fact, is inherited from pairwise error probability (PEP) between two codewords as shown in Fig.. According to Definition, let i =, both clusters C(ū ) and C(ũ ) contain only one codeword and it is obtained that P [C(ū ) C(ũ ) c ] = P [c c ] = P [W (y c ) < W (y c )], which indicates that the CPEP is reduced to a PEP when i =. As shown in Fig. (a), a codeword pairwise error means that the received vector y is closer to the distorted codeword c than the transmitted codeword c, measured by Euclidean distance, and the PEP P [c c ] denotes the probability that the transmitted codeword c is decided as the given distorted codeword c. Similarly, as shown in Fig. (b), a cluster pairwise error could be intuitively understood as that the received vector y is closer to the codewords within the error cluster C(ũ i ) in an average sense, and the CPEP P [C(ū i ) C(ũ i ) c ] denotes the probability that the transmitted codeword c (c C(ū i )) is decided as from C(ũ i ) rather than from C(ū i ). (3)

5 5 Assuming the binary input AWG (BiAWG) channel in this paper, we only need to consider the all-zero codeword in the evaluation of error performance, i.e., only the CPEP P [C(0 i ) C(ũ i ) 0 ] is considered in this paper, where ũ i 0 i. Given a polar code (, M, A), the transmitted codeword 0 C(0 i ), the received vector y and an error path ũ i, the path metrics of the correct path 0 i and the error path ũ i are obtained as (y, û i =0i u i =0) = (y, û i =ũi u i =ũ i ) = u i+ {0,} i W (y (0 i, u i+)g ), (4) u i+ {0,} i W (y (ũ i, u i+)g ). (5) According to Definition, (3), (4) and (5), it is obtained that P [C(0 i ) C(ũ i ) 0 ] = P W (y c ) < W (y c ) c C(0i ) c C(ũi [ ) ] = P (y, û i =0i (i) 0)<W (y, û i =ũi ũ i). (6) Before proceeding to the calculation of (6), let us look at a special case of CPEP P [C(0 i ) C(ũ i ) 0 ] with the error path ũ i =(0 i, ). Using (6), one obtains P [C(0 i ) C(0 i, ) 0 ] = P [ (y, û i = P [ =0i log (y, û i =0i 0) (i) 0) < W (y, û i ] (y, ûi =0i ) < 0 =0i )]. (7) It is clear that the CPEP P [C(0 i ) C(0 i, ) 0 ] is the decision error probability over the bit channel derived in [9], where SC decoder is assumed. It is shown in [9] that the upper bound on the FER of a polar code is given as i A P [C(0i ) C(0 i, ) 0 ], where A is the unfrozen bit index set. Later in [0] [], the calculation methods with low computation complexity for P [C(0 i ) C(0 i, ) 0 ] are proposed, such as Tal-Vardy s method [0] or Gaussian approximation [], []. However, there is no work regarding the CPEP calculation of the general case, i.e., CPEP P [C(0 i ) C(ũ i ) 0 ], where the methods in [9] [] can not be directly applied. IV. CALCULATIO OF CPEP In this section, the calculation of CPEP P [C(0 i ) C(ũ i ) 0 ] with any given error path ũ i 0 i is provided. According to (6), one obtains that P [C(0 i ) C(ũ i ) 0 ] = P [L (i) (y, 0 i, ũ i )<0], (8) where the log-likelyhood ratio (LLR) L (i) (y, 0 i, ũ i ) is defined as L (i) (y, 0 i, ũ i )=log (y, û i =0i 0) (y, ûi =ũi ũ i). (9) In the following subsections, we present the recursive relationship between (y, 0 j, ũj ) and L(j ) (y, 0 j, ũj ). Based on this relationship, the distribution of L (i) (y, 0 i, ũ i ) and the probability P [L (i) (y, 0 i, ũ i )<0] could be obtained in a recursive manner. A. Recursive Calculation of P [C(0 i ) C(ũ i ) 0 ] According to (), it is obtained that W (j ) (y, ũ j ũ j ) =W (j) (y, ũ j W (j ) (y, 0 j 0) =W (j) (y, 0 j )+W (j) )+W (j) (y, ũ j 0), (0) (y, 0 j 0). () Using (9)-(), the recursive relationship between (y, 0 j, ũj ) and L(j ) (y, 0 j, ũj ) is derived as (y, 0 j, ũj { ) = (y, 0 j L (j ), ũj ) log(+e LC )+log(+e LE ), ũ j =0 L (j ) (y, 0 j, ũj ) log(+e LC )+log(+e LE ), ũ j =, () where LC and LE are LLRs given as below, LC:= (j) [y, 0 j, W (0j, )]=log LE:= [y,(ũ j,0),(ũj (j) W,)]=log (y, û j =0 j 0) W (j) (y, ûj =0 j ), (y,û j =ũ j 0) W (j) (y,ûj =ũ j ). Given an error path ũ i, the LLR L (i) (y, 0 i, ũ i ) could be calculated recursively as (), in which the recursive calculation of (y, 0 j, ũj ) starts from L(i0) (y, 0 i0, ũi0 ), and ends at L (i) (y, 0 i, ũ i ), where i 0 denotes the index of the first error bit in ũ i, i.e., ũ i0 =(0i0, ). Among the LLRs involved in (), the distributions of L (i0) (y, 0 i0, ũi0 ) and [y, 0 j, (0j, )] (i 0 <j i)(i.e., LC), could be efficiently obtained as [0] [], as mentioned at the end of Section III. As for [y, (ũ j, 0), (ũj, )] (i 0 < j i)(i.e., LE), its distribution will be detailed later in Section IV-B. Intuitively, if one could recursively calculate the probability density function (pdf) of (y, 0 j, ũj ) (denoted by f j(x)) from pdf of L (j ) (y, 0 j, ũ j ) (i.e., f j (x)) based on (), as the density evolution in [7], the CPEP could be obtained easily by P [C(0 i ) C(ũ i ) 0 ] = 0 f i(x) dx. Unfortunately, due to the complex interaction in (), we are unable to derive a closed-form expression of recursive pdf calculation based on (). Therefore, we resort to a Monte Carlo method to obtain the CPEP P [C(0 i ) C(ũ i ) 0 ], directly using the recursion ().

6 6 In the Monte Carlo method, we start from i 0 by randomly generating CPEP samples for L (i0) (y, 0 i0, ũi0 ), L(i0+) [y, 0 i0+, (0 i0, )](i.e., LC) and L (i0+) [y, (ũ i0, 0), (ũi0, )](i.e., LE), according to their distributions. Using (), we obtain the CPEP samples of L (i0+) (y, 0 i0+, ũ i0+ ), and store these samples in the cache. Afterwards, the CPEP samples for L (i0+) [y, 0 i0+, (0 i0+, )](i.e., LC) and L (i0+) [y, (ũ i0+, 0), (ũ i0+, )](i.e., LE) are randomly generated according to their distributions, and using the already obtained CPEP samples of L (i0+) (y, 0 i0+, ũ i0+ ) in the cache, we obtain the CPEP samples of L (i0+) (y, 0 i0+, ũ i0+ ) by (), and stores them in the cache. In this way, the recursion continues until j = i and the CPEP samples of the target LLR L (i) (y, 0 i, ũ i ) are obtained. Finally, the CPEP P [C(0 i ) C(ũ i ) 0 ] is determined as P [C(0 i ) C(ũ i ) 0 ]=P [L (i) (y, 0 i, ũ i )<0] = (the number of samples of L (i) (y, 0 i, ũ i )<0)/ CPEP. The CPEP-calculation method described above is summarized as Algorithm. Algorithm : The Calculation of CPEP P [C(0 i ) C(ũ i ) 0 ], (ũ i 0 i ) Input:, σ, CPEP, the error path ũ i, i 0 (the index of the first error bit in ũ i ). : for j = i 0 to i do : if j = i 0 then 3: Obtain the distribution of L (i0) (y, 0 i0, ũi0 ) as [], [], and randomly generate CPEP samples for L (i0) (y, 0 i0, ũi0 ), based on its distribution. 4: else 5: Obtain the distribution of [y, 0 j, (0j, )](i.e., LC) as [], [], and randomly generate CPEP samples for [y, 0 j, (0j, )], based on its distribution. 6: Obtain the distribution of [y, (ũ j, 0), (ũj, )](i.e., LE) as mentioned in Section IV-B, and randomly generate CPEP samples for [y, (ũ j, 0), (ũj, )], based on its distribution. 7: Obtain the CPEP samples of (y, 0 j, ũj ) based on (). 8: end if 9: end for 0: Obtain the number of samples of L (i) (y, 0 i, ũ i ) < 0, and calculate the CPEP as P [C(0 i ) C(ũ i ) 0 ] = P [L (i) (y, 0 i, ũ i ) < 0] = (the number of samples of L (i) (y, 0 i, ũ i )<0)/ CPEP. Output: the CPEP P [C(0 i ) C(ũ i ) 0 ]. B. Distribution of [y, (ũ j, 0), (ũj, )] As [], [9] and [], the LLR [y, (ũ j, 0), (ũj, )] (i.e., LE in ()) could be recursively calculated as L (k ) [y, (ũ k, 0), (ũ k, )] =tanh {tanh( L(k) / [y/,(ũ k,e ũk,o,0),(ũk,e ũk,o,)] ) tanh( L(k) / [y /+, (ũk,e, 0), (ũk,e, )] )}, (3) L (k) [y, (ũ k, 0), (ũ k, )] = L (k) / [y /+, (ũk,e, 0), (ũk,e, )] +( )ũk L (k) / [y/, (ũ k,e ũk,o, 0), (ũk,e ũk,o, )], (4) for k =,,, /, where ũ k,e and ũ k,o are the subvectors of ũ k with even and odd indices, respectively, and y is the received vector of the codeword 0. Based on (3) and (4), we are able to apply the Gaussian approximation to recursively calculate the pdfs of L (k ) and L (k) from those of L(k) /, where =, 4,,, and L () (y t, 0, ) = log W (yt 0) W (y ( t ) σ, 4 σ ), ( t ), where σ denotes the noise variance of a BiAWG channel, and bits 0, are mapped into,, respectively. The basic idea to obtain the distribution of [y, (ũ j, 0), (ũj, )] is to approximate it and the other LLRs involved in (3) and (4) as Gaussian random variables. This method is named as Gaussian approximation, and is firstly used to simplify the density evolution under AWG channel in LDPC codes [8], and later used in polar codes [], [], since the recursions in (3) and (4) are essentially the same as those of belief propagation decoding in LDPC codes. However, it is noted that the symmetry condition in [8] is generally not satisfied in our case, i.e., we can not assume V { [y, (ũ j, 0), (ũj, )]} = E{, 0), (ũj, )]}, where V { } and E{ } [y, (ũ j denote the variance and mean of a random variable, respectively. The reason is that, different from [], [], ũ j is not assumed to be all zeros in our case. Therefore, different from Gaussian approximation in [8], which only evaluates the means, we have to evaluate both the means and variances of the LLRs in (3) and (4) in order to obtain the distribution of [y, (ũ j, 0), (ũj, )]. Also due to the unsatisfied symmetry condition, we are unable to derive a closed-form expression for recursive calculation of the means and variances of L (k ) and L(k) from L(k) / as [], []. Therefore, we again resort to a Monte Carlo method to calculate the means and variances, i.e., the distributions, of L (k ) and L(k) based on (3) and (4). In the Monte Carlo calculation, the samples of input LLRs L (k) / [y/, (ũ k,e ũk,o, 0), (ũk,e ũk,o, )] and L (k) / [y /+, (ũk,e, 0), (ũk,e, )] in (3) and (4) are randomly generated for a given number LE, according to their distributions. Then, one calculates LE samples of output LLRs L (k ) [y, (ũ k, 0), (ũ k, )] and

7 7 () (5) (3) (7) L y,( u,0),( u,) () (6) (4) (8) (4) 4 4 L ( y,00,0) () L ( y,000,00) L ( y,00,0) L ( y,000,00) (4) () 4 3 L ( y,00,0) () 6 5 L ( y,00,0) () 8 7 Fig.. The LLRs transformation process with = 8. () L 4 ( y,0,)~ (, ) () L 4 ( y,0,)~ (, ) () L 4 ( y3,0,)~ (, ) () L 4 ( y4,0,)~ (, ) () L 4 ( y5,0,)~ (, ) () L 4 ( y6,0,)~ (, ) () L 4 ( y7,0,)~ (, ) () L 4 ( y8,0,)~ (, ) i.e., LE in (), which involves a number of Monte Carlo distribution calculations based on (3) and (4), due to the lack of an analytical calculation on this distribution. Each of the Monte Carlo distribution calculation involves the generation of LE samples for each LLR on the right of (3) or (4), and LE calculations of (3) or (4). Calculation of distribution of [y, (ũ j, 0), (ũj, )] involves the recusive calculation of the distributions of LLRs L (k ) and L(k) from those of L (k) /, for =, 4,,, which incurs /, /4,, Monte Carlo distribution calculations, repectively. These add up to Monte Carlo distribution calculations of (3) or (4). Then, the calculation of CPEP P [C(0 i ) C(ũ i ) 0 ] incurs the complexity of (i i 0 )( ) Monte Carlo distribution calculations of (3) or (4) in total. Apparently, this is a significant computational burden, especially when the code length is large. L (k) [y, (ũ k, 0), (ũ k, )] using (3) and (4). Afterwards, the means and variances of output LLRs are es- samples. In this way, one sequen- timated by their LE tially obtains the distributions of LLRs L (k ) from those of L (k) /, where [y, (ũ j, 0), (ũj and L(k) =, 4,,, to get, )] in a recursive manner, from the initial distribution L () (y t, 0, ) ( σ, 4 σ ), ( t ). An example obtaining the mean and variance of the LLR [y, (ũ j, 0), (ũj, )] is presented as follows, Example : Given a polar code (, M, A) = (8, 4, {4, 6, 7, 8}), this example shows how to obtain the distribution of the LLR L (7) 8 [y8, (ũ 6, 0), (ũ 6, )], where the error path ũ 6 = 00000, and y 8 denotes the received vector of the transmitted codeword 0 8. The noise variance of the BiAWG channel is σ and the bits 0, are mapped into,, respectively. In Fig., the dashed lines show the transformation process from the eight initial LLRs L () (y t, 0, ) ( σ, 4 σ ), ( t 8) to the LLR L (7) 8 [y8, (ũ 6, 0), (ũ 6, )], which is a reflection of the recursions (3) and (4). The first step is to estimate the means and variances of the four LLRs L () (y, 00, 0), L () (y4 3, 00, 0), L () (y6 5, 00, 0) and L () (y8 7, 00, 0) according to (4). Let us take L () (y, 00, 0) as an example, one can randomly generate the samples of L () (y, 0, ) and L () (y, 0, ) for a given number LE, according to L () (y, 0, ) ( σ, 4 σ ) and L () (y, 0, ) ( σ, 4 σ ). Then, the LE samples of L () (y, 00, 0) are calculated as L () (y, 00, 0) = L () (y, 0, ) + L () (y, 0, ). Afterwards, the mean and variance of L () (y, 00, 0) are estimated by its LE samples. Following the same way, one further obtains the distributions of L (4) 4 (y4, 000, 00), L (4) 4 (y8 5, 000, 00) and finally L (7) 8 [y8, (ũ 6, 0), (ũ 6, )]. C. Complexity of the Proposed CPEP Calculation Method From the CPEP calculation described above, it is not difficult to see that the major complexity comes from the calculation of the distribution of [y, (ũ j, 0), (ũj, )], D. An Example of CPEP Calculation Results In this subsection, we show the sum of CPEPs ũ P [C(0 i ) C(ũ i ) 0 i 0i ], where ũ i represents legal error path, through an example, and compare it to the accumulated elimination error probability i k= P (B k). Example : In this example, we firstly compare the sum of CPEPs obtained by Algorithm and by computer simulations. The codeword length is set as = 3, and the number of information bits is set as M = 0. The parameters and M are chosen such that the total paths is 04, which is a number we can manage in the calculation and simulation. The information bit index set is A={6, 3, 4, 6, 7, 8, 9, 30, 3, 3}, the noise variance of the BiAWG channel is σ =, and the transmitted codeword is 0 3. Fig. 3 shows the sum of CPEPs, which is given as ũ P [C(0 i ) C(ũ i ) 0 3 i 0i ] for i A, where ũ i represents legal error path. The parameter CPEP in Algorithm is set as CPEP = 0 6, the parameter LE in Section IV-B is set as LE = In computer simulations of the summed CPEPs, the number of trials for each P [C(0 i ) C(ũ i ) 0 3 ] is 0 6. It is observed in Fig. 3 that the results of Algorithm match the simulation results well. Another simulation with this example compares the sum of CPEPs and the accumulated elimination error probabilities i k= P (B k), for i A, as shown in Fig. 4. Since the event B i represents that the correct path is eliminated exactly at the bit u i during the SCL decoding, the accumulated elimination error probability i k= P (B k) denotes the probability that the correct path is eliminated between u and u i, and P (B k )=0, if k is a frozen bit index. In simulation of Fig. 4, we obtain i k= P (B k) by computer simulations with list sizes L =,, 4, 8, and the number of trials for each P (B k ) is 0 6. For comparisons, the accumulated elimination error probabilities i k= P (B k) with L =,, 4, 8 are normalized, such that the maximum normalized values are equal to the maximum sum of CPEPs. It is observed that, the five curves in Fig. 4 are well correlated, although not accurate. Therefore, the sum of CPEPs could be used as an indicator of the accumulated elimination error probability, e.g., it could be used to predict

8 Computer simulations Proposed calculation method parity function parity function 0.4 Sum of CPEPs Bit channel index (i) Fig. 3. The comparison between the sum of CPEPs obtained by Algorithm and by computer simulations. Probability Sum of CPEPs i ( P(B k )).00,L= k= i ( P(B k )) 5.94,L= k= i ( P(B k )) 8.65,L=4 k= i ( P(B k )) 64.70,L=8 k= Bit channel index (i) Fig. 4. The comparison between the sum of CPEPs and the accumulated elimination error probabilities. at which bit channels the elimination error would occur with high probability, in an SCL decoder. V. PCC POLAR CODE COSTRUCTIO USIG CPEP In this section, the conventional parity-check-concatenated (PCC) polar code is introduced, and the gain of error performance due to concatenation of parity check code is explained from the perspective of CPEP. Moreover, we apply the CPEP tool to optimize the parity functions of the PCC polar code, to reduce elimination errors of the SCL decoder. A. PCC Polar Code The PCC polar code proposed in [5] is a concatenation scheme of a polar code with an outer parity-check code. Specifically, the encoding bit sequence u at the polar encoder in a PCC polar code contains information bits, frozen bits and parity bits. As shown in Fig. 5, parity bits are scattered in u, and each parity bit together with its preceding information bits u u u 3 Information bit Frozen bit Fig. 5. The parity functions in PCC polar codes. Parity bit forms a parity function. In more detail, a PCC polar code could be defined as a 4-tuple (, I, P, {T k k =,,, K}), where is the codeword length, I {,,, } is the index set of information bits in u, P {,,, } is the index set of K parity bits in u, {T k k =,,, K} contains K subsets of I, i.e., T k I, and the elements in each subset T k are the information bit indexes corresponding to the k-th parity function. The K parity bits are calculated as u pk = ( i T k u i ) mod, k =,,, K, (5) where p k is the k-th element in P and u pk is the k-th parity bit in u. For PCC polar codes, the k-th parity bit index p k should be larger than the information bit indexes in the k-th parity function, i.e., p k > max(t k ), where max(t k ) denotes the largest element in T k. In a PCC polar code, the unfrozen bit sequence, containing information bits and parity bits, is given by A = I P, and the frozen bit sequence is given by A c = {,,, } \ A. With the information bit sequence u I = (u i, i I), the parity bits u P = (u i, i P) and the frozen bits u A c = (u i, i A c ), the encoding sequence u is determined and the concatenated codeword is generated by c =u G. At the receiver, the received vector y is fed into a parity-check-aided SCL decoder, where the information bits are decided by the path metrics [], the parity bits are decided from the decided information bits using their corresponding parity functions [5], and the final output is the most likely path in the list []. It is demonstrated in [5] that, with proper parity functions, the correct path is more likely to be reserved in the list, i.e., elimination errors are reduced. However, the reason of the performance gain was not adequately explained and [5] fails to present any criterion based on which construction of the parity functions could be optimized. With CPEP, we are able to explain the performance gain brought by the parity bits, and further give an optimization criterion for the construction of parity functions. From the CPEP perspective, properly designed parity bits can lower the CPEP better than frozen bits do, thus making the correct path more competitive and reducing elimination errors in SCL decoding with limited list size. Among u of a PCC polar code, each non-information bit acts as either a frozen bit, or a parity bit. If there exists a parity function that gives smaller CPEPs than a frozen bit does, replacing the frozen bit with a parity bit would result in certain performance gain. In another words, minimizing the sum of CPEPs could be used as a criterion in searching of the good parity functions for PCC polar code. u

9 9 In the followings, we will present the construction of the PCC polar code in detail, which mainly considers to reduce elimination errors by optimizing the parity functions. As for picking errors, CRC code could be used to remove most of them []. An example of this is the short polar encoder for control channels in 5G technical specification [], which combine the PCC polar code with a CRC code to reduce both elimination errors and picking errors, and more details about this combination are discussed in Section VI. B. A Greedy Optimization for {T k k =,,, K} Based on CPEP As explained in the previous subsection, construction of a PCC polar code is equivalent to determination of {T k k =,,, K}. In this subsection, we propose a greedy optimization of T k, k =,,, K, at the aim of minimizing the sum of CPEPs ũ p k 0 p k P [C(0 p k ) C(ũp k ) 0 ] at the k-th parity bit u pk, i.e., p T k =arg min P [C(0 k ) C(ũp k ) 0 ], T k ũ p k U(T,,T k,t k ) (6) where U(T,, T k, T k ) denotes a set containing all legal error paths at the k-th parity bit, given that {T k k =,,, k } has been determined and the k-th parity function is T k. A legal error path ũ p k in U(T,, T k, T k ) should satisfy that ũ p k 0p k, and 0 or, j {,,, p k } I 0, j {,,, p k } A c ũ j = ( ũ i ) mod, j =p t and t=,,, k i T t ( ũ i ) mod, j =p k i T k (7) for j =,,, p k. It is clear from (6) that the K parity functions are sequentially constructed from T to T K, i.e., with the already constructed T t (t=,,, k ), the k-th parity function T k is optimized as (6), so that the competitiveness of the correct path is maximized against the error paths at the k-th parity bit. Since each parity function is optimized based on the former ones, we call (6) a greedy optimization. Two complexity issues need to be addressed regarding the optimization: i) the number of candidate T k at u p k is Ik, which could be extremely large even for codes of moderate length, where I k denotes a set containing the information bits before the k-th parity bit, i.e., I k =I {,,, p k }. ii) the number of error paths at u pk is Ik, which also could be unacceptable complexity. In the following subsection, approaches to narrow down the search scope of T k are proposed to address the first issue. In Subsection V-D, a PCC polar code construction with top-j-cpep list is proposed to address the second issue. C. Approaches to arrow Down the Search Scope of T k In this subsection, two basic ideas that narrows down the search scope of T k in (6) are proposed: ) restricting the elements of T k within a subset of I k, and ) restricting the dimension of T k, such as T k =. In the first approach, we restrict that T k Ĩk, where Ĩk I k and Ĩk = I, where I is the number of selected elements from I k. Therefore, the search complexity of T k is reduced from O( Ik ) to O( I ). Intuitively, the bits in I k with the I largest error probabilities are chosen as the elements of Ĩk, here, the error probability is the decision error probability over the corresponding bit channel [9]. After the set Ĩk is obtained, the search scope of T k is given as S k = { T k T k Ĩk}. (8) In the second approach, we restrict the dimension of T k as T k =. Then, the search scope of T k is generated as S k = {{I k ()}, {I k ()},, {I k ( I k )}}, (9) where I k (i) is the j-th element of I k. Therefore, the search complexity of T k in (6) is reduced from O( Ik ) to O( I k ). The restriction T k = means each parity-check code is degraded into a repetition code. With this approach, the PCC polar code is named as repetition-concatenated polar code. After the search scope S k is obtained as (8) or (9), the optimization (6) is rewritten as T k = arg min T k S k p P [C(0 k ) C(ũp k ) 0 ]. U(T,,T k,t k ) ũ p k D. PCC Polar Code Construction with Top-J-CPEP List (0) Computing the CPEPs P [C(0 p k ) C(ũp k ) 0 ] of all error paths in U(T,, T k, T k ) could be a huge amount of complexity, since the number of error paths is Ik. To address this complexity issue, we adopt a list of J error paths, called as top-j-cpep list, and carry out the optimization of (6) only over these error paths, instead of all error paths. In this way, the number of error paths to calculate the sum of CPEPs in (6) is reduced from Ik to J. The paths in the top-j-cpep list are extended from u to the K-th parity bit u pk, in a progressive manner. Denote by u i,l (l =,,, J) the J error paths in the top-j-cpep list after decision of u i, then each path u i,l is extended from u i to u i as follows. If i I (u i is an information bit), J + error paths are obtained, among which J paths are obtained by splitting each path u i,l into two error subpaths, i.e., ( u i,l, 0) and ( ui,l, ), and one path is (0i, ). Comparing these J + paths, J error paths with the largest CPEPs P [C(0 i ) C(ũ i ) 0 ] are reserved in the list. If i A c (u i is a frozen bit), each error path u i,l is directly extended as ( u i,l, 0). If i P (u i is the k-th parity bit u pk ), with the k-th parity function T k optimized as (), each path u i,l is extended as ( u i,l, u i =( j T k u j,l ) mod ).

10 0 M v Combined CRC and CRC u ˆM I c PCC Polar Encoder Channel PC aided SCL Encoder Decoder CRC-PCC Polar Encoder y v Information bit Parity bit Frozen bit 0 Fig. 6. Block diagram of encoding and decoding of a CRC-PCC polar code. Fig. 7. []. Parity functions implemented by a five-length cyclic shift register With the top-j-cpep list, the optimization of T k using is give by T k = arg min T k S k J l= P [C(0 p k ) C( up k,l ) 0 ]. () Finally, the PCC polar code construction algorithm is summarized as Algorithm. Apparently, complexity of this algorithm is no larger than J CPEP calculations. Algorithm : The PCC Polar Code Construction Using CPEP Input:, I, P, K, J, σ. : for i = to p K do : if i=p k (k =,,, K) then 3: Obtain S k (the search scope of T k ) as (8) or (9). 4: Determine T k as (). 5: Extend each path u i,l in the top-j-cpep list into ( u i,l, u i =( u j,l ) mod ). j T k 6: else if i I then 7: Denote the number of error paths in the top-j-cpep list as L, and split each path u i,l (l =,,, L ) in the list into two error sub-paths ( u i,l, 0) and ( u i,l, ), plus the error path (0i, ), L + error paths are obtained. 8: If L + J, all L + error paths are reserved in the list. Otherwise, J error paths with the largest CPEPs are reserved. 9: else 0: Extend each path u i,l in the top-j-cpep list into ( u i,l, 0). : end if : end for Output: the constructed PCC polar code (, I, P, {T k k =,,, K}). VI. SIMULATIO RESULTS In this section, we first introduce the combined CRC and PCC polar code (CRC-PCC polar code), constructed using the proposed method in Section V, and then present the simulation results on the frame error rate (FER) and undetectable error rate (UER) performance. A. Combined CRC and PCC Polar Code and Simulation Parameters The CRC-PCC polar code is a combination of a CRC code and a PCC polar code, as shown in Fig. 6. The information bit sequence v M is firstly encoded by an outer CRC code of length L CRC. Then, the CRC codeword is encoded by a PCC polar encoder to generate the final codeword. The CRC codeword is viewed as part of the information bit sequence in the PCC polar code (, I, P, {T k k =,,, K}), i.e., the index set I contains M original information bits plus L CRC CRC bits. At the receiver, the decisions of information bits, parity bits and frozen bits are carried out as the parity-check-aided SCL decoder [5], and the final output is the most likely CRC-valid path if there is at least one CRC-valid path in the list []. The CRC-PCC polar code is an effective combination to reduce both elimination errors and picking errors. Specifically, the elimination errors could be reduced through the optimized parity checks, and the picking errors could be ignored with the aid of a long CRC code, as pointed out in [9], [5]. Due to the advantages of CRC-PCC polar codes, CRC-PCC polar code has been adopted as one of the channel encoders in 5G technical specification []. For the CRC-PCC polar codes in the simulation, the CRC length is set as L CRC = 9, and its generator polynomial is g(x) = x 9 +x 6 +x 4 +x 3 +x +x 0 +x 8 +x 7 +x 4 +x 3 + [0]. Indexes of M + L CRC bit channels with the smallest decision error probabilities are chosen as the information bit set I. Then, the parity bit set P is chosen as the indexes of K bit channels with the smallest decision error probabilities among the bit channels that have indexes greater than the first information bit channel index and are different from the information bit channel indexes. If we keep no frozen bit channel after the first information bit channel, i.e., all the bit channels that have indexes larger than the first information bit channel index are set as either information or parity check bit channels, the code is called as Full-Check CRC-PCC polar code. During the construction of parity functions, the parameters LE in Section IV-B and CPEP in Algorithm are set as LE =5 0 4 and CPEP = 0 6, respectively. The parameter I in Section V-C is set as I = 0, then the dimension of S k (the search scope of the parity function T k ) is S k = I = 0 =04. The number of error paths in the top-j-cpep list in Algorithm is set as J =7. For comparison, we also simulated the CRC-PCC polar code with the parity functions standardized in 5G technical specification [], which are implemented by a five-length cyclic shift register [] as shown in Fig. 7. The basic idea of the design is: the information bits are sequentially input into a five-length cyclic shift register, and the check result stored in the leftmost register is output as parity bits, i.e., u pk, when needed. In the simulations, the BPSK-AWG channel model is

11 FER CRC Polar CRC PCC Polar,standardized parity functions CRC PCC Polar,CPEP construction R /3 R=/ R / Eb/0(dB) Fig. 8. The FER performance of the CRC-PCC polar codes with = 5 and R = /3, /, /3. FER R=/3 CRC Polar,Block CRC PCC Polar,Block,standardized parity functions CRC PCC Polar,Block,CPEP construction R=/ R=/3 CRC Polar,Known Bit CRC PCC Polar,Known Bit,standardized parity functions CRC PCC Polar,Known Bit,CPEP construction Eb/0(dB) Fig. 9. The FER performance of the punctured CRC-PCC polar codes with M = 40, R = /3, /, /3, known-bit puncturing and block puncturing. employed with the noise variance σ = R E b / 0, where E b / 0 is the signal-to-noise ratio, R=M/ or R=M/ p is the code rate, M is the number of information bits, is the unpunctured codeword length, and p is the punctured codeword length. When calculating the decision error probabilities over the bit channels, the E b / 0 is set such that the FER is nearly 0 3 [0]. Finally, the list size of the SCL decoder is set as L=8. B. FER Performance of CRC-PCC Polar Codes ) CRC-PCC Polar Codes without puncturing Fig. 8 compares the FER performance of the CRC-PCC polar codes and CRC-concatenated polar codes with codeword length = 5 and code rates R = 7/5 /3, R = / and R = 34/5 /3. Compared to the CRC-concatenated polar code, the FER performance gain of the CRC-PCC polar codes with three different code rates is nearly 0.dB at FER of 0 3. Moreover, the standardized parity functions are able to achieve nearly the same error performance as the parity functions constructed with CPEP do, in these cases. ) Punctured CRC-PCC Polar Codes With puncturing, the elimination errors are more likely to occur, due to the increased number of non-fully polarized bit channels. Fig. 9 compares the FER performance of the punctured CRC-PCC polar codes and CRC-concatenated polar codes with the number of information bits M =40, the code rates R = /3, /, /3, and two puncturing patterns named as known-bit puncturing [3] and block puncturing. In knownbit puncturing, the bits with indexes { p +, p +,, } in c are punctured, and the encoding bits with indexes {BitRev( p, )+, BitRev( p +, )+,, BitRev(, ) + } in u are set as frozen bits (zeros), where p denotes the codeword length after puncturing, denotes the mother codeword length, and BitRev(i, ) denotes the bitreversal map of i, where the length of the binary representation of i is log, for example, BitRev(6, 8) = 3. It can be verified that the punctured bits are always zeros in knownbit puncturing [3], i.e., they are known to the receiver. In block puncturing, the bits with indexes {,,, p } are punctured. Fig. 9 shows that the CRC-PCC polar codes outperform the CRC-concatenated polar codes over various codeword lengths, code rates and puncturing patterns. The largest performance gain is nearly 0.5dB at FER = 0 3, with M = 40, R = /3, /3, and block puncturing. Besides, the differences in FER performance between the constructed and standardized parity functions are relatively small. An interesting observation from Fig. 9 is that while the known-bit puncturing shows superior performance over the block puncturing for CRC-concatenated polar codes, it is not always the case for the CRC-PCC polar code. The CRC-PCC polar codes with block puncturing works better than those with known-bit puncturing, when the number of punctured bits is large (see the cases with R=/3, /3). This difference could be explained by the fact that the number of available parity bits with block puncturing is generally more than that with knownbit puncturing, especially when the number of punctured bits is large. With block puncturing, the parity bit index set is P = {min(i), min(i)+,, } \ I, for a full-check CRC- PCC polar code. However, with known-bit puncturing, certain bit channels with low decision error probabilities have to be set as frozen bit channels, in order to ensure the punctured bits are known to the receiver [3]. Then, the encoding bits with indexes {BitRev( p, ) +, BitRev( p +, ) +,, BitRev(, )+} should be excluded from parity bit index set, which makes less number of available parity bits for known-bit puncturing. Since the elimination errors are more likely to be reduced with a large number of parity bits, the performance gain achieved by CRC-PCC polar codes with block puncturing could be more significant than that with known-bit puncturing. 3) Repetition-Concatenated Polar Code