On Efficient Decoding of Polar Codes with Large Kernels

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1 On Efficient Decoding of Poar Codes with Large Kernes Sarit Buzago, Arman Fazei, Pau H. Siege, Veeresh Taranai, andaexander Vardy University of Caifornia San Diego, La Joa, CA 92093, USA {sbuzago, afazeic, psiege, vtaranai, Abstract Defined through a certain 2 2 matrix caed Arikan s kerne, poar codes are known to achieve the symmetric capacity of binary-input discrete memoryess channes under the successive canceation (SC) decoder. Yet, for short bockengths, poar codes fai to deiver a compeing performance under the ow compexity SC decoding scheme. Recent studies provide evidence for improved performance when Arikan s kerne is repaced with arger kernes that have smaer scaing exponents. However,for kernes the time compexity of the SC decoding increases by a factor of 2. In this paper we study a specia type of kernes caed permuted kernes. The advantage of these kernes is that the SC decoder for the corresponding poar codes can be viewed as a permuted version of the SC decoder for the conventiona poar codes that are ined through Arikan s kerne. This permuted successive canceation (PSC) decoder outputs its decisions on the input bits according to a permuted order of their indices. We introduce an efficient PSC decoding agorithm and show simuations for two permuted kernes that have better scaing exponents than Arikan s kerne. I. INTRODUCTION Introduced by Arikan [1], poar codes are the first codes that were proved to achieve the symmetric capacity of a binary-input discrete memoryess channe (B-DMC) W. Poar codes can be viewed as part of a much arger famiy of codes that are generated according to the 2 2 matrix F 2 ( ). These ength n 2 m codes are cosets of a inear subspace that is spanned by some k rows of F2 m,themth Kronecker power of F 2. One important exampe of such codes are Reed- Muer codes, for which the spanning k rows correspond to the k rows of F m 2 that have maximum Hamming weight. The genius of Arikan s construction ies in the fact that the seection of the k spanning rows depends on the channe. Each row of F m 2 is associated with a synthesized channe caed a bit channe, which is ined through F 2 and the channe W. The seected k rows correspond to the k east noisy bit channes, where the noisiness of the channe is measured by its Bhattacharyya parameter (see [1] for more information on the Bhattacharyya parameter). A remarkabe property of F 2 is that as n grows, the bit channes exhibit a poarization phenomenon: some of the bit channes become very noisy, whie the rest of the bit channes become amost noiseess. Moreover, the fraction of channes that are amost noiseess approaches the symmetric capacity of the channe, I(W). As a resut, for every R<I(W) and for sufficienty arge n there exists a rate R poar code for which the probabiity of error under a successive canceation (SC) decoder is O(n 1 2+ε ), where ε>0 is arbitrariy sma. Despite their impressive asymptotic behavior, empirica resuts indicate ess impressive performance of the SC decoder for poar codes of short bock-engths, e.g. compared to LDPC codes. To improve performance, different decoders were suggested, e.g., beief propagation decoding [2], [5], [8], decoding with ook-ahead [15], and ist successive canceation decoding [12] improved SC decoding [4]. Combined with cycic redundancy check (CRC) pre-coding, the ist successive canceation decoder provides the best performance for poar codes up-to date. Meanwhie, a series of works pursued the same goa by repacing F 2 with arger matrices caed kernes that may induce a faster poarization of the bit channes [6], [7], [9], [10], [11], [14]. To estimate how poarizing a kerne is, two figures of merit were proposed. The first is the error exponent [3], [10] which quantifies the exponentia decay of the error probabiity with respect to the bock-ength n, for a fixed rate R. The second is the scaing exponent [6], [9] which refects the dependence between the ength of the code and its rate, for a fixed probabiity of error P e. More precisey, a scaing exponent μ is a constant that depends ony on the kerne K and the channe W for which n Θ(I(W) R) μ, where the sum of the Bhattacharyya parameters of the nr east noisy bit channes is at most P e. No method is known to compute the scaing exponent for genera channes. In fact, it is not even cear whether or not the scaing exponent exists. Hassani et. a. [9] introduced a numerica method to compute the scaing exponent for the binary erasure channe (BEC) and found that μ 3.627, where the kerne is F 2. Fazei and Vardy [6] presented an 8 8 kerne K 8 with μ and showed that this is optima for kernes of size 8. They aso suggested a heuristic construction through which they found a kerne K 16 with μ However, the time compexity of a SC decoding of poar codes with kernes scaes as 2 n og n. The goa of this paper is to achieve both good performance and efficient SC decoding. To this end we propose to use a specia type of kernes caed permuted kernes. These kernes are formed by permuting the rows of F2. One exampe of a permuted kerne is the kerne K 8 from [6]. On the other hand, the kerne K 16 ined in [6] is not a permuted kerne. Whie a successive canceation decoder for a poar code with the kerne F 2 and of dimension k decides on the n input bits u 0 u 1...u n 1 (n uncoded bits, n k of them are known to the decoder) one after the other according to the sequentia order /17/$ IEEE

2 from 0 to n 1, a SC decoder for poar codes with permuted kernes decides on the input bits according to a permuted order of their indices. Therefore, we ca the SC decoder for a poar code with a permuted kerne a permuted successive canceation (PSC) decoder. By expoiting the connection, we introduce a more efficient impementation of the PSC decoder. Due to page imitations, we ony describe our PSC decoding agorithm for ength- poar codes. A more detaied presentation of the agorithm wi be given in the fu version of this paper. We aso propose two new permuted kernes and show simuation resuts for their performance. The rest of this paper is organized as foows. In Section II, we present notation and initions that are used throughout the paper and review the basic concepts of poar codes. In Section III we discuss the connection between the bit between permuted kernes and F 2 channes of a permuted kerne and the bit channes of F 2. Our PSC decoding agorithm for bock-ength decoder is presented in Section IV. Time compexity of the agorithm is discussed in Section V aong with some simuation resuts. II. PRELIMINARIES In this section we introduce notation and initions used throughout the paper and review the basic concepts for poar codes. For a positive integer n, denote by [n] the set of n integers {0, 1,...,n 1}. For two integers a<b, denote by [a, b] the set of b a integers {a, a +1,...,b 1}. For a positive integer and for a r [], denote by [n] r the set of a eements in [n] that are equa to r moduo, where shoud be cear from the context. A binary vector of ength n is denoted by u n 1 0 u 0 u 1...u n 1.ForA [n], denote by u A the subvector of u n 1 0 that is specified according to indices from A. In particuar, if divides n and r [] then u [n]r u r u +r...u n +r. A operations on vectors and matrices in this paper are carried out over the fied GF (2). The componentwise addition moduo-2 of two binary vectors u n 1 0 and v0 n 1 is denoted by u n 1 0 v0 n 1. For an matrix K, denote by K m the mth Kronecker power of K. Throughout this paper W : X Y is a generic B- DMC with input aphabet X {0, 1}, output aphabet Y, and transition probabiities W(y x), where for a x X and y Y, W(y x) is the conditiona probabiity that the channe output is y given that the transmitted input is x. For a positive integer n, denote by W n : X n Y n the channe that corresponds to transmission over n independent copies of W. Hence, for every x n 1 0 X n and y0 n 1 Y n,the transition probabiity W n (y n 1 0 x n 1 0 ) is given by W n (y0 n 1 x n 1 0 ) n 1 W(y i x i ). i0 For an n n matrix G, aseta [n] of size k, and a vector z0 n k 1,etC be the code that encodes a ength n input vector u n 1 0,forwhichu [n] A z0 n k 1, to the codeword x n 1 0 u0 n 1 G. Ifi [n] A, then u i is caed a frozen bit. For a i [n], theith bit channe with respect to G, G : X Y n X i, is ined by the transition probabiities G (yn 1 0,u i 1 0 u i ) u n 1 i+1 X n i n 1 W G(y n 1 0 u n 1 0 G). (1) A successive canceation (SC) decoder for C outputs a decision vector ˆu n 1 0 in n steps, where at the ith step the decoder decides on the vaue of ˆu i according to the foowing rue. If i [n] A then ˆu i is set to the vaue of the frozen bit u i. Otherwise, the decoder cacuates the pair of probabiities P i [0] G (yn 1 0, ˆu i 1 0 u i 0), (2) P i [1] G (yn 1 0, ˆu i 1 0 u i 1). and sets ˆu i to the more ikey vaue according to these probabiities, i.e., { 0, Pi [0] P ˆu i i [1] 1, otherwise. Notice that, in genera, the compexity of the SC decoder may be exponentia in n, since the cacuation of P i [0],P i [1] requires a summation of 2 n i 1 terms. Let n 2 m and et G n B n F2 m, where B n is the bitreversa permutation matrix (see [1] for the inition of B n ). For G G n, a poar code with Arikan s kerne is ined by setting A to be the set of k indices corresponding to the bit-channes with the owest Bhattacharyya parameters 1. Due to the structure of G n,thesc decoder for poar codes admits an efficient impementation that runs in O(n og n). This foows from the fact that for this choice of G, there is a recursive expression for the transition probabiities of the ith bit channe. That is, for a i [n], ifi 2φ then G n (y0 n 1,u i 1 0 u i ) 1 2 W(φ) G n/2 (y n/2 1 0,u [2φ]0 u [2φ]1 u 2φ u 2φ+1 ) u 2φ+1 W (i/2) G n/2 (y n 1 n/2,u [2φ] 1 u 2φ+1 ), and if i 2φ +1 then G n (y0 n 1,u i 1 0 u i ) 1 2 W(φ) G n/2 (y n/2 1 0,u [2φ]0 u [2φ]1 u 2φ u 2φ+1 ) W ( i/2 ) G n/2 (y n 1 n/2,u [2φ] 1 u 2φ+1 ). W. Notice that, for r {0, 1}, [2φ] r is the set of a eements in [2π] that are equa to r moduo 2. A kerne K is an invertibe matrix. As mentioned above, one can improve the performance of poar codes by repacing Arikan s kerne with arger kernes that admit better W (0) G 1 1 By the inition of poar codes, the vaues of the frozen bits are aso required. If the channe is symmetric, then the frozen bits are a taken to be zero. For asymmetric channes, an assignment of the frozen bits that guarantees a vanishing probabiity of error is known to exist, however no practica method that finds such an assignment is known. (3) (4)

3 scaing exponents, and hence are considered to be more poarizing. Next, we describe the matrix G G n,k through which a poar code with kerne K is generated. Let n m and et R n be the permutation matrix for which u0 n 1 R n u [n]0 u [n]1...u [n] 1, for a u n 1 0 X n. For an kerne K (K r,s ), ine the matrix G n,k recursivey by G,K K and G n,k (I n/ K)R n (I G n/,k ). For ease of notation we denote the ith bit channe with respect to G n,k by n,k. The recursive structure of the matrix G n,k induces recursive formuas for the bit channes as foows. Lemma 1. Let K be an kerne that can be obtained by permuting the rows and coumns of some ower trianguar matrix. For a i [n], ifi φ + ρ for some φ [n/] and ρ [] then n,k (yn 1 0,u i 1 0 u i ) W (φ) n/,k (y(s+1)n/ 1 sn/ u (φ+1) 1 φ+ρ+1 1 s0,t (s) (u [φ] ) r0 1 K r,s u φ+r ), where T (s) (u [φ] ) r0 1 K r,s u [φ]r. Notice that u [φ]r is a vector of ength φ and K r,s GF (2), hence T (s) (u [φ] ) is a vector of ength φ as required by the inition of the φth bit channe. The term r0 1 K r,s u φ+r is simpy the inner product of u (φ+1) 1 φ with the sth coumn of K. Aso notice that for 2,the expression in Lemma 1 reduces to the recursive formuas (3) and (4). From Lemma 1, it foows that a SC decoding agorithm at the kerne eve, i.e., for a ength- poar code with G K, that has time compexity t can be extended to a SC decoding agorithm for a ength-n poar code with kerne K that has time compexity O(tn og n). In particuar, there exists an impementation for the SC decoder that runs in O(2 n og n). In practice, this time compexity may be too arge even for reativey sma vaues of. For this reason we propose to use a specia type of kernes that can simutaneousy reduce the time compexity of the SC decoder and achieve better scaing exponents. These kernes are caed permuted kernes since they are ined by a row permutation of the matrix G B F2 L, 2 L. III. PERMUTED KERNELS In this section, we focus our discussion on poar codes of ength 2 L, where the matrix G is an permuted kerne. We wi express the bit channes of these codes in terms of the bit channes of G. The discussion in this section motivates the SC decoding agorithm that is formaized in Section IV. A permutation π of ength is a bijection π :[] []. For a permutation π, the matrix permutation corresponding to π is denoted by M π and ined by (M π ) r,s { 1, if s π(r) 0, otherwise, i.e., u 1 0 M π v 1 0, where v π(r) u r, for a r []. A permuted kerne with respect to π is ined by M π G M π B F2 L. Let C π be the poar code that is generated by G K π. For i [], etd i {π(0),π(1),...,π(i 1),π(i)} and et d i max{d i }. For ease of notation, for a i [n] we denote the ith bit channe with respect to G n,kπ and G n by W n,π (i) and W n (i), respectivey. K π Lemma 2. For a i [], theith bit channe with respect to K π is equa to,π (y 1 0,u i 1 0 u i ) W (di) (y0 1,v di 1 0 v di ), v [di +1] D i where v0 1 u0 1 M π, i.e., for a r [], v π(r) u r. Proof. By inition of the ith bit channe (1), we have that,k (y 1 0,u i 1 0 u i ) u 1 i u 1 i W (y 1 0 u 1 0 K π ) W (y 1 0 (u 1 0 M π )G ) v [] Di W (y 1 v [di +1] D i W (di) v [di +1] D i v 1 d i +1 0 v 1 0 G ) W () (y 1 0 v 1 0 G ) (y 1 0,v di 1 0 v di ). By Lemma 2, the ith bit channe with respect to K π can be expressed in terms of the d i th bit channe with respect to G. In particuar, using a SC decoder for the ength- poar code with Arikan s kerne, one can ine a SC decoder for the code C π as foows. For a i [], to decode ˆu i,given the previous decoded bits ˆu i 1 0, the transition probabiities W (di) (y0 1,v di 1 0 v di ) are cacuated for a possibe choices of v s, s [d i +1] D i, where v sπ(r) ˆu r if r [i]. Thus, 2 di+1 i transition probabiities for the ith bit channe of G are cacuated in order to cacuate the ith pair of probabiities ined in (2). IV. FORMALIZATION OF SC DECODER FOR PERMUTED KERNELS In this section we formaize our SC decoder for a ength- code C π ined by a permuted kerne K π. As mentioned above, SC decoding for C π is equivaent to SC decoding of a ength- poar code with kerne F 2 that decides on the input bits in a permuted order according to π. For this reason, we ca our SC decoding scheme permuted successive canceation (PSC) decoding. We present an impementation of PSC decoding for any permutation π, which requires significanty ess computationa power compared to the conventiona SC decoder at the kerne eve. Anaysis of the time compexity is aso presented.

4 For i [], etˆu i 1 0 be the bits decoded so far. Reca that in the ith step, the SC decoder cacuates the pair of probabiities ined in (2). Denote these probabiities by 16, 16 For v i 0,et Q i [0],π (y 1 0, ˆu i 1 0 u i 0), Q i [1],π (y 1 0, ˆu i 1 0 u i 1). P i [v0] i (y0 1,v0 i 1 v i ). Reca that D i {π(0),π(1),...,π(i)} and d i is the maximum over D i.thedecoding window in the ith step is denoted by DW i [d i +1] D i. By Lemma 2, Q i [b] P di [v di 0 ], (5) v s: s DW i where v π(i) u i b and for a r [i], v π(r) ˆu r. The PSC decoding utiizes the conventiona recursive SC decoder for the cacuation of P di [v di 0 ] with different settings (corresponding to a choices for v s, s DW i ) so that it can generate the desired pair of probabiities Q i [b], b {0, 1}. The decoding window DW π (i), refers to the indices of the input bits in v di 0 that are not decoded yet, and hence have to be set temporariy before execution of an instance of the traditiona SC decoder. However, the origina (and efficient) recursive agorithm requires that {π(0), π(1),, π(i)} {0, 1,,i}, which may not be the case for many i []. To overcome this probem, the SC decoder must aso cacuate P j [v j 0 ] for a j [τ i +1,d i +1], where τ i { min(dw i ) 1, i 1, if DW i otherwise. In other words, τ i denotes the argest index j such that v 0,v 1,,v j have a been decoded before cacuating the pair of probabiities associated with u i v π(i). To better understand the importance of this parameter, we reca that the conventiona SC decoding agorithm requires memory, in which it stores both cacuated probabiities and decoded vaues associated with the previous bit channes and bit channes in inner-ayer nodes. Utiizing this memory, the agorithm avoids dupicate cacuations and achieves O(n og(n)) overa compexity in decoding a of the bit channes. This od information, i.e., probabiities and decoded vaues of the previous bit channes, is ony hepfu if it can be propagated back through the poarization ayers and revea unknown inner nodes. However, that is ony possibe when the indices of the aready-decoded bits form a consecutive set. The parameter τ i determines the argest subset of consecutive indices of the aready-decoded bits and hence acts as the starting point for the recursive SC decoding. For simpicity, we ca the sets [τ i +1] and [τ i +1,d i +1] the decoding tai and decoding ookahead of u i, respectivey. Note that, depending on π, forsomei [] we may have DW i, which is equivaent to {π(0),,π(i)} {0, 1,,i} and hence ony a singe instance of the traditiona SC decoder is required. Fig. 1: Structure of the permuted kerne K π from Exampe 1. The recursive structure of G 16 is expoited by the PSC decoder for this kerne. Exampe 1. Consider the permuted binary kerne K π G 16,π, shown in Figure 1, where π (0, 1, 2, 4, 8, 3, 5, 6, 9, 12, 7, 11, 13, 14, 15). The scaing exponent of this kerne for BEC is μ(k π ) To decode u 0,u 1, and u 2 we can simpy ca the traditiona recursive SC decoder. However, to decode u 3, one needs to cacuate the pair of probabiities Q 3 [b] P 4 [ˆv 0ˆv 1ˆv 2 0b]+P 4 [ˆv 0ˆv 1ˆv 2 1b], (6) where ˆv π(r) denotes the decoded bit ˆu r. By inition, we have that D 3 {0, 1, 2, 4}, DW 3 {3}, and τ 3 2. Each term in the right-hand side can be cacuated using the recursive SC decoder with input parameters y0 15, ˆv 0,v and y0 1, ˆv 0,v However, in order to cacuate P 4 [ˆv 0ˆv 1ˆv 2 v 3 v 4 ], the recursive SC must first cacuate P 3 [ˆv 0ˆv 1 ˆv 2 v 3 ]. Moving forward, to decode the next unknown bit u 4,itis desired to output the probabiity pair Q 4 [b] P 8 [ˆv 0ˆv 1ˆv 2 v 3ˆv 4 v5b] 7 (7) v 3v 5v 6v 7 which consists of terms. In this case, D 4 {0, 1, 2, 4, 8}, DW 4 {3, 5, 6, 7}, and τ 3 2. Again, in order to cacuate P 8 [ˆv 0ˆv 1ˆv 2 v 3ˆv 4 v5b], 7 the recursive SC must first cacuate P j [ˆv 0ˆv 1 ˆv 2 v j 3 ], for a j [τ 3 +1, 7]. Surprisingy, the decoding of u 5 formuation to (7), as Q 5 [b] foows a very simiar v 5v 6v 7 P 8 [ˆv 0ˆv 1ˆv 2 bˆv 4 v 7 5ˆv 8 ]. (8)

5 and enters a nested chain of for-oops which aternate between a 2 DWi initiaizations for the unknown bits in the decoding window of u i. At each such initiaization, the agorithm runs the traditiona successive canceation for a j [τ i +1,d i +1] with input parameters y0 1, ˆv j 1 0, where s [j], ˆv s is given either from the previousy decoded vaues ˆu i 1 0, or the temporary initiaization of the eements in the decoding window. It is observed that both the size of the decoding window and the size of the decoding ookahead have a significant roe in determining the time compexity of the decoding agorithm. In the next section, we present the simuations resuts for mutipe permutations, aong with a more detaied comparison on the compexity increase for each permutation. V. SIMULATION RESULTS Fig. 2: Structure of the agorithm that cacuates the transition probabiities of the ith bit channe,π (y 1 0, ˆu i 1 0 u i) for PSC decoding. Accordingy, u 5 can be decoded by simuating the conventiona SC at the same bit channe 2. An overview of the genera PSC decoder is given in Figure 2. The diagram is sef-expanatory. We add a few comments here to carify some of the instructions. The reader is encouraged to read the fu version of the paper for the detaied decoding agorithm (simiar to source code) at both the kerne and the fu-ength eve, where more detais about efficient memory addressing, and dupicationfree cacuations are presented. The agorithm starts by checking updates on the decoding tai. The decoding tai (and its propagated vaues) correspond to the fixed inner-ayer nodes in the decoder structure, and hence by storing their corresponding probabiities/hardvaues in memory we can avoid dupicate cacuations. Next, the agorithm moves to the first unknown node, i.e., τ i, 2 It is possibe to expoit this fact by storing a of the terms in (7) in an auxiiary memory to avoid dupicate cacuations in the future stages (decoding u 5,u 6, and u 7 ). We refer the reader to the fu version of the paper, where we expain this extension in detai, its improvement in throughput, and the extra required memory. In this section, we discuss the compexity increase in more detai and compare the performance of the poar codes based upon different permuted kernes with respect to their overa decoding compexity. As expained earier, the computationa compexity of the permuted SC decoder for the ith bit channe in the permuted kerne may be more than the corresponding ith bit channe in the traditiona kerne. This is mainy because computing Q i [b] requires a summation over 2 DWi terms, where each term can be computed by running d i τ i instances of the conventiona SC decoder; see (8). However, not a such instances consume the same amount of computationa power. Hence, to arrive at an expicit formuation of the computationa compexity for the proposed agorithm, we need to anayze each instance separatey. It is to be noted that time compexity and space compexity are the two main notions of computationa compexity. This paper focuses on improving the decoder speed (throughput) and ignores the efficiency of the memory management. In the fu version, we propose agorithms that sacrifice space efficiency for utimate throughput, or vice versa. The speed of a successive canceation decoder is usuay imited to the number of mathematica operations that have to be executed sequentiay. This sequence in the conventiona SC decoder consists of a doube visit for each of n og(n) nodes in the decoder structure; the first visit is to output a pair of probabiities for the nodes on their eft (coser to u i s), and the second visit is to cacuate and update the probabiities for the nodes on the right side (coser to y i s). Let us ine the normaized compexity fraction at bit channe i, ρ i,tobe the fraction of the tota 2n og(n) computationa power that is spent on decoding W i,gn. It is easy to track the number of nodes contributing to reveaing each v i, and hence a simpe program can derive the expicit vaues of ρ i s. Tabe I shows the evauation of ρ i s for K π from Exampe 1. i odd {2, 6, 10, 14} {4, 12} {8} {0} ρ i 0.008% 0.039% 0.101% 0.226% 0.351% TABLE I: Tabe of normaized compexity fractions for 4 th power or Arikan s kerne. ρ i 2 16 og(16) the number of visits (out of 2 16 og(16)) the reguar SC decoder makes on the inner nodes whie decoding the ith bit channe.

6 As expained in the previous section, the PSC decoder executes the instances of traditiona SC at each index i mutipe times. Let us ine the frequencies, f i, by the tota number of cas made by PSC on traditiona SC at index i. The foowing theorem gives an expicit formua for the computationa compexity of PSC decoder at the kerne eve. Theorem 1. The normaized computationa compexity of the PSC decoder at kerne eve, CC(K,π ), is given by 1 CC(K,π ) f i ρ i. (9) i0 The proof is straightforward and thus it is omitted. Note that the overa decoding compexity at the fu-ength aso scaes with the same scaar as in (9). Tabe II tabuates the evauation of f i s for the permuted kerne in Exampe 1. Utiizing (9), we have CC(K 16,π )7.03. i f i i f i TABLE II: The frequency tabe for K π from Exampe 1. Exampe 2. Consider a different permuted binary kerne K σ, ined by the permutation σ (0, 1, 2, 3, 4, 6, 8, 10, 5, 9, 7, 11, 12, 13, 14, 15). The scaing exponent of K σ for the binary erasure channe is given by μ 3.541, which is again smaer than μ(f 2 ). Hence one woud expect a performance improvement upon conventiona poar codes. However, since σ(i) i, for a i [15] {5, 6, 7, 8, 9, 10}, it is aso expected to observe a smaer increase in computation compexity. We can verify this by foowing the formuation in (9), and show CC(K σ ) 3.2. We eave the discussion on space compexity to the fu version of the paper, where we aso introduce even faster decoding agorithms which can increase the throughput even further at the cost of extra memory. Figure 3 depicts the performance comparison of three poar codes at ength n 2 8 over binary erasure channes. Two of these codes are constructed via K σ and K π, respectivey, and the other one is the conventiona Arikan s poar code, constructed from the 2 2 kerne F 2. It is observed that the poar code constructed from K π, and decoded with PSC, outperforms both the conventiona poar code and the code constructed by K σ. The atter outperforms the conventiona poar code. The resuts are in agreement with the convention that the smaer the scaing exponent, the better the poar code performance. Notice that for some erasure channes the poar code constructed from K π outperforms Arikan s poar code even with the ML decoder. It is to be noted that the conventiona construction agorithms such as [13] cannot be appied directy for the kernes with 4 due to the exponentia increase in the number of bit channe outputs. In this paper, we utiized a Monte- Caro construction agorithm, which may be improved by Frame Error Rate (FER) SC Decoder for F 2 Permuted SC Decoder for K π Permuted SC Decoder for K σ ML Bound for F Erasure Probabiity Fig. 3: Frame-error-rate comparison of three poar codes over binary erasure channes. The tree codes are constructed with F 2, K σ,and K π, respectivey, at code ength n 2 8, and decoded with SC and PSC agorithms. A codes are optimized for the channe BEC(0.2) and rate R 3. The ML bound on performance of the F2 poar 5 code is aso given, based upon methods in [12]. using a arger number of iterations. We eave the study of ow-compexity construction agorithms for future work. REFERENCES [1] E. Arikan, Channe poarization: A method for constructing capacityachieving codes for symmetric binary-input memoryess channes, IEEE Trans. on Inform. Theory, vo. 55, pp , Juy [2] E. Arikan, A performance comparison of poar codes and Reed-Muer codes, IEEE Commun. Letters, vo.12, pp , June [3] E. Arikan and E. Teatar, On the rate of channe poarization, Proc. IEEE Int. Symp. Inform. Theory, pp , Seaou, South Korea, Juy [4] K. Chen, N. Kai, and L. Jiaru, Improved successive canceation decoding of poar codes, IEEE Trans. on Commun., vo. 61, pp , August [5] U. U. Fayyaz and J. R. 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