Sequential Decoding of Polar Codes with Arbitrary Binary Kernel

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1 Sequentia Decoding of Poar Codes with Arbitrary Binary Kerne Vera Miosavskaya, Peter Trifonov Saint-Petersburg State Poytechnic University Emai: Abstract The probem of efficient soft-decision decoding of poar codes with any binary kerne is considered. The proposed approach represents a generaization of the sequentia decoding agorithm introduced recenty for the case of poar codes with Arikan kerne. Numeric resuts show that the proposed agorithm enabes near-ml decoding of poar codes with BCH kerne. I. INTRODUCTION Poar codes were shown to be abe to achieve the capacity of a wide cass of communication channe 1. Namey, it was shown that m-fod appication ( of a 2 2 inear transformation 1 given by matrix A = (Arikan kerne transforms the 1 1 origina memoryess binary input output-symmetric channe into a number of bit channe, and their capacities converge with m to and 1. However, the rate of convergence (rate of poarization appears to be quite ow. It was shown in 2 that high-dimensiona kerne (e.g. based on BCH codes provide higher poarization rate than the Arikan kerne. That is, the decoding error probabiity of such poar codes decreases much faster with code ength compared to simiar Arikan codes. However, there are sti no efficient methods for their decoding. The successive canceation (SC agorithm is the cassica decoding method for poar codes. However, it is not abe to recover from errors which may occur at eary phases of the decoding, and fai therefore to achieve the ML performance. This probem was addressed in 3, where a ist decoding agorithm for Arikan poar codes was introduced. It was shown in 4, 5 that the same performance can be achieved with much smaer compexity by empoying stack decoding agorithm. It was shown in 6 that further compexity reduction can be obtained at the expense of negigibe performance degradation. In this paper we present a generaization of the sequentia decoding agorithm suggested in 6 to the case of poar codes with an arbitrary binary kerne. The paper is organized as foows. Background on poar codes and their sequentia decoding is provided in Section II. The proposed decoding method is described in Section III. Some improvements for it are derived in Section IV, and impementation issues are discussed in Section V. Numeric resuts are provided in Section VI. A. Poar codes II. BACKGROUND An (n = m,k poar code based on kerne B is a inear bock code generated by k rows of matrix G n = M (n B m, where B m denotes m-times Kronecker product of matrix with itsef and M (n is permutation matrix, such that M (n t,t = 1 for any t = m 1 i= t i i and t = m 1 i= t m 1 i i, t i,..., 1}. Let c = (c,...,c. Any codeword of a poar code can be represented as c = u G n, where u is the input sequence consisting of k information symbo and n k frozen symbo, which are equa to zero. Let F be the set of n k indices of frozen symbo. B. Successive canceation decoding The decoding probem for poar codes consists in finding û = arg P(u u :u,f = y, where u,f denotes subvector of u consisting of eements with index F. At the i-th phase the SC decoder computes estimates arg ui P(û i 1,u i y, t F û i = (1, otherwise. Given some path u i, its probabiity can be recursivey computed as P(u +t y = 1 π, (2 where t <, π = P θ B, = r = (θ B u (s+1 1, y (+1 n 1 n ( u (r+1 1 r G n, r s, P(u r y r = P(yr urp(ur P(y r, and P(y r u r is the channe transition probabiity function. The SC agorithm for binary poar codes can be impemented with compexityo(nog (n unit cacuations, where a unit cacuation corresponds to singe evauation of (2. C. Sequentia decoding of poar codes with Arikan kerne A maor drawback of the SC agorithm is that it cannot correct errors which may occur at eary phases of the decoding process. This probem is soved in stack/ist agorithms by keeping a ist of the most probabe paths within code tree 3, 4, 5. A path of ength i is identified by vaues u i 1,1} i. Each path is associated with its score, which depends on its probabiity. Stack agorithms 4, 5 keep the paths in a stack (priority queue. At each iteration the decoder seects

2 for extension path u i 1 with the argest score, and performs the i-th phase of SC decoding. That is, if i F the path is extended to obtain u i, where u i =, and the extended path is stored in the stack together with its score. Otherwise, the path is coned to obtain new paths (u,...,u i 1, and (u,...,u i 1,1, which are stored in the stack together with their scores. In order to keep the size of the stack imited, paths with ow scores can be purged from the stack, so that the tota number of paths considered simutaneousy by the decoder does not exceed some parameter Θ. Furthermore, if the decoder returns to phase i more than L times, a paths shorter than i + 1 are eiminated. Decoding terminates as soon as path of ength n appears at the top of the stack, or the stack becomes empty. Hence, the worst case compexity of stack decoding is given by O(Lnog (n unit cacuations. Average decoding compexity depends on how path scores are defined. In 6 a ow-compexity version of the stack agorithm for binary poar codes with 2 2 Arikan kerne was introduced. Let v, < 2 n i 1, be different paths such that v i = ui and v i+1,1} n i 1. Let J be a random variabe, which is equa to,...,2 n i 1 1 } if the most probabe codeword of the poar code corresponds to path v. The soution of the optimization probem Z = (u y is equa u :u,f = Y to Y (v y with probabiityp J = }, i.e. it is a function of random variabe J. Here U h is a random variabe corresponding to the h-th input symbo of poarizing transformation and Y h is a random variabe corresponding to the h-th received symbo. Observe that J = impies that v h =,h F. Hence, one can estimate Z as E J Y (vj y. It can be seen that increasing i reduces the number of possibe vaues of random variabe J, improving thus the accuracy of such estimate. Let us assume without oss of generaity that v is the most probabe path, i.e. arg <2 n i 1 Y (v y =. Event J = is equivaent to v h =,h F. Therefore, one obtains E J 2 n i 1 1 = Y Y Y (vj y = (v y P J = } (v y P J = }. }} R(u i,y It is possibe to show that R(u i,y = u i+1 Y (u y (3 can be computed exacty as 6 R(u 2s+t,y = 1 u 2(s+1 1 2s+t+1 = R (θ A u 2(s+1 1,,y (+1n 2 1, n 2 where t, 1}, and initia vaues for these recursive expressions are given by R(b,y = P(b y, b,1}. It is difficut to compute the second term in (3 exacty for any given y at phase i of the SC decoder. However, it can be averaged over a possibe received sequences. Average probabiity of the most probabe path u having prefix u i, having zeroes in positions F, is ower bounded by probabiity of the SC decoder, which starts from u i and does not take into account any freezing constraints, makes decisions u =, > i, F, i.e. does not make errors in these positions. Probabiity of this event is given by ˆΩ(i = F,>i (4 (1 P, (5 wherep is the-th subchanne error probabiity, provided that exact vaues of a previous bits u, <, are avaiabe. It depends ony onn, F (i.e. the code being considered, channe properties and phase i. For any given channe, probabiitiesp can be pre-computed using density evoution. Thus, the score for path u i can be defined as ˆT(u i,y = R(u i,y ˆΩ(i. (6 This approach enabes one to compare paths u i with different engths, and prevent the decoder from switching frequenty between different paths. A. Decoding agorithm III. PROPOSED APPROACH We propose to generaize decoding agorithm 6 to the case of poar codes based on arbitrary binary kerne. The idea of this agorithm does not use any kerne properties. Simiary to the case of Arikan kerne, in the case of kerne B one obtains R(u +t,y = 1 = R (θ B,,y (+1n 1, n where t <. This enabes one to perform decoding of a poar code with arbitrary binary kerne in exacty the same way as in the case of Arikan kerne, i.e. keep paths u i in a stack, and seect at each iteration for extension a path with the highest vaue of ˆT(u i,y, unti a path of ength n is obtained. However, the task of computing (7 is not as simpe as in (4. Moreover, efficient techniques for computing ˆΩ(i shoud be provided. (7

3 B. Computing path score 1 FunctionR(u i,y : Leti = +t, t <. Finding imizing (7 is equivaent to decoding in a coset of (,κ = t 1 code C t, which is generated by ast κ rows of matrixb. The coset is given byu +t B..t +C t, whereb..t is the submatrix of B consisting of first t + 1 rows. Indeed, task (7 can be considered as finding pu +t t = P( z 1, (8 where( transition probabiities P(b z = (θb R u 1,,b,y (+1n 1, b,1}, <, n and are some scaing factors. This enabes one to empoy any soft decision decoding agorithm for finding imizing (8. Hence, the tota compexity of the sequentia decoding agorithm is O(Lnog (n unit operations, where unit operations correspond to searching for the most probabe codeword in codes generated by rows of kerne B. 2 Function ˆΩ(i: Finding error probabiity P, F, for a bit subchanne induced by non-arikan poarizing transformation, provided that exact vaues of u 1 are avaiabe to the decoder, sti remains an open probem. Therefore, we have to use simuations to compute these vaues. Possibe aternatives incude empoying union bound together with mutieve code weight enumerator computation techniques introduced in 7, and Gaussian approximation method based on Gram-Charier series expansion of the expression for symbo LLR 8. The first approach appears to work poory on bad bit subchanne, which are incuded into the set of frozen symbo. The second approach may resut in divergent series. IV. IMPROVED DECODING METHOD It can be seen that estimation of input symbo u i, s i ( < (s + 1, is performed using probabiities R u (θb 1,,b,y (+1 n 1,b,1}, n and the corresponding subset of frozen symbo is F s,...,s+ 1}. The former probabiities are independent from t = i mod. Therefore, symbo u (s+1 1 s can be estimated ointy. This enabes one to significanty reduce the decoding compexity. Assume that at some iteration β of the decoding agorithm described in Section III-A path u s 1 is seected for extension as the most probabe one. Let us define the set of paths of ength (s+1 which can be obtained from path u s 1 } V = u (s+1 1 u (s+1 1 s,1},u (s+1 1 s,f =. Reca, that the sequentia decoding agorithm invoves stack purging operation, which eiminates paths with owest scores, so that the tota number of paths does not exceed Θ, and ist pruning operation, which ensures that at most L paths can be extended ti phase i for any i. Let us assume for the sake of simpicity that a paths given by V are inserted into the stack, and then stack purging and ist pruning operations are appied. Let Φ V be the set of paths which survive these operations. (s+1 1 ˆT(u,y It can be seen from (5 (6 that scores have common factor ˆΩ((s + 1 1, i.e. Φ consists of φ = Φ paths u (s+1 1 with argest probabiities R(u (s+1 1,y. This impies that these paths correspond to φ most probabe codewords u (s+1 1 s B of (,λ s code Υ s generated by rows of B with indices from Ξ s, where Ξ s = s,s+1,...,(s+1 1}\F is the set of non-frozen symbo in the s-th bock of input symbo, and λ s = Ξ s. These codewords can be identified using any ist soft decision decoding agorithm for binary inear code Υ s. Let ˆt be the -th argest path score stored in the stack at iteration β, < Θ. If there are Θ < Θ active paths in the stack, assume that ˆt =, Θ < Θ. Let Q i be the set of paths of ength i stored in the stack. Let t be the scores of paths in Q (s+1 arranged in the descending order. Let t =, L < L, where L = Q (s+1. Φ incudes a paths from V which can survive stack purging and ist pruning operations, i.e. u (s+1 1 (s+1 1 V : ˆT(u,y (ˆt Θ φ, t L φ. (9 Observe that if L L 2 λ and Θ Θ 2 λs, then exhaustive search over codewords w 1 Υ s is performed and for each w 1 corresponding path is pushed into the stack together with its score. It appears that set Φ given by (9 is highy redundant. More efficient agorithm can be obtained by repacing Φ with } ˆΦ = u (s+1 1 Φ ˆT(u (s+1 1,y e α ρ, (1 where ρ = T ( (u s,w 1,y, and w 1 B is the most probabe codeword of Υ s. Here α is a parameter which affects the size of the obtained ist. It must be recognized that for sufficienty high α this approach may require decoding of Υ s beyond its Johnson bound 9. This may resut in ist size, which is exponentia in. In order to obtain a practica agorithm, it may be necessary to further restrict ˆΦ to contain L vectors with the argest score for some L. The improved decoding agorithm operates as foows. At each iteration it seects for extension path u s 1 with the argest score. Then ist decoding of code Υ s is performed, and paths given by (1 are pushed into the stack together with their scores. The input data for the ist decoder are ( (θb u 1,,b,y (+1 n 1 n sti given by R, b,1}. These vaues are given by (7, and are computed recursivey by empoying a soft-input hard-output decoder for codes generated by rows of B. Path score is defined as where ˆT(u (s+1 1,y = R(u (s+1 1,y Ω(s+1, ˆΩ(s = n/ 1 σ=s+1 (1 Γ σ, and Γ σ is the probabiity of non-zero vaues appearing in positions σ,...,(σ +1 1} F of the most probabe path with prefix u σ 1. Stack purging and ist pruning

4 operations are performed in the same way as described above. Probabiities Γ σ can be obtained from simuations. Decoding terminates as soon as path of ength n appears at the top of the stack. V. IMPLEMENTATION Both encoding and decoding operations for a poar code of ength n = m can be decomposed into m ayers, where each ayer corresponds to appication of the inear transformation given by B to m 1 data units obtained at the previous ayer. In the case of decoding, ayer m 1 (fina ayer corresponds to recovery of input symbo u i, where ist decoding is needed, as described in Section IV. At ayers,...,m 2 (intermediate ayers one needs to identify ust singe most probabe codeword given by (8 for each data unit. A. Intermediate ayers Any soft decision decoding agorithm for coset u +t B..t + C t can be used to find a soution of (8. Let us consider for the case of concreteness the case of Box and Match agorithm 1. For a code of dimension κ, (κ+ψ,γ Box and Match agorithm performs search for up to 2γ errors in positions from set Λ (t Ψ (t, where Λ (t is the most reiabe information set and Ψ (t is the set of ψ most reiabe positions except positions in Λ (t. Parameter ψ specifies a trade-off between time and space compexity. Letωb = P(b+u +t B..t, z be the input probabiities. The input LLR vector χ 1 is defined as χ = ( 1 u+t B..t, og ω1, ω where B..t, is the vector consisting of first t + 1 eements of the -th coumn of matrix B. In order to identify the most probabe codeword vt+1 1 B t C t one needs to construct sets Λ (t and Ψ (t. The first step of the Box and Match agorithm consists in sorting of the reiabiity vector ( χ,..., χ 1. It can be seen that the permutation obtained as a resut of this step does not depend on t, i.e. it can be performed ony once for any given s. Observe that Λ ( Λ (1 Λ ( 2. In order to reduce the compexity, we propose to construct information sets Λ (t ointy. Assuming that C 2 is a repetition code, one obtains that Λ ( 2 = 2 } : 2 = arg χ. Given Λ (t, one can construct Λ (t 1 = Λ (t t 1 }, where t 1 is such that B t.. 1,t 1 is ineary independent of B t.. 1,i,i Λ (t, and χ t 1 is as high as possibe. Ψ (t can be obtained simiary. Given the output vt+1 1 B t+1.. 1of the decoding agorithm, one shoud compute pv t t = 1 = ω(v 1 B. Even if vt+1 1 B t fai to be the ML soution of the decoding probem due to suboptimaity of the decoder being used, this does not necessariy resut in a faiure of the proposed poar code decoding agorithm, since pv t t is typicay cose to its true vaue. Reca that for any t one needs to compute both p t and p1 t. This requires finding most probabe codewords c 1 and c1 1 in cosets (v t 1,B..t +C t and (v t 1,1B..t + FER FER Θ=124, 6 ML ower bound Poar Arikan with CRC, L= E b /N, db Figure 1. Performance of (124, 512 poar codes 1-4 Θ=124, 6 Θ=372, L=32 Poar Arikan with CRC, L= E b /N, db Figure 2. Performance of (496, 248 poar codes C t, respectivey. Observe that the most probabe codeword c 1 in coset v t 1 B..t 1 +C t 1 has aready been computed at step t 1. Obviousy, c 1 or c1 1 is equa to c 1, so at the t-th step it remains to invoke decoder for C t ony once. It can be seen that the compexity of computing R(u s+t,y,u +t,1}, t <, for intermediate ayers of the decoder is given by 2 t= N(, t 1,ψ,γ+ O(og+O( 3, wheren(,κ,ψ,γ is the compexity of the (κ+ψ,γ Box and Match agorithm for (,κ code, excuding sorting and information set construction steps. B. Fina ayer At the ast ayer one needs to construct a ist of codewords given by (1. This can be impemented using either ordered statistics 11 or a ist extension of the Box and Match agorithm 12. Observe that one does not need to perform any cacuations, incuding any processing at intermediate ayers, whie decoding fuy frozen bocks of input symbo.

5 Average number of iterations Average number of iterations Θ=124, E b /N, db Figure Average decoding compexity for (124, 512 poar code Θ=124, 6 Θ=372, L= E b /N, db Figure 4. Average decoding compexity for (496, 248 poar code VI. NUMERIC RESULTS Figure 1 iustrates the performance of (124, 512, 16 poar code 1 with BCH kerne 2 and = 32 for the case of BPSK transmission over AWGN channe. For comparison, the resuts for Arikan poar code concatenated with CRC and ist decoding (see 3 is ao shown. It can be seen that for sufficienty arge L the proposed decoding agorithm enabes one to achieve near-ml performance. At ow SNR poar codes with BCH kerne outperform Arikan poar code concatenated with CRC. However, at high SNR the performance of the former one becomes imited by poor minimum distance. Figure 2 provides simiar resuts for the case of (496, 248, 32 poar code with BCH kerne and = 64. Athough minimum distance is sti quite ow, error foor does not appear down to error probabiity 1 5. It is possibe to improve the performance of poar code with BCH kerne by empoying dynamic subchanne freezing techniques introduced in 5. Figures 3 4 iustrate the average number of iterations performed by the proposed decoding agorithm for the considered 1 This poar code was constructed using the method given in 13, and has codes Υ s, s < n/, with dimensions,,,,,,,,, 1, 1, 5, 7, 11, 13, 16, 21, 21, 24, 26, 26, 28, 29, 31, 31, 31, 31, 31, 32, 32, 32, 32. codes. Ony iterations corresponding to non-empty bocks of non-frozen symbo are counted. It can be seen that at sufficienty high SNR, where codeword error probabiity drops beow 1 3, the proposed agorithm requires in average t iterations, where t n/ is the number of non-empty bocks Ξ s. VII. CONCLUSIONS In this paper a nove decoding agorithm for binary poar codes with arbitrary kerne was proposed. The agorithm is a generaization of the sequentia decoding agorithm introduced earier for the case of poar codes with Arikan kerne. It invoves near-ml soft-decision decoding of codes generated by rows of submatrices of the kerne. Simuation resuts show that the proposed approach enabes one to perform near-ml decoding of poar codes with BCH kerne. These codes were shown to outperform Arikan poar codes concatenated in CRC, at east in the ow-snr region. The proposed agorithm can be used for decoding of poar codes with dynamic frozen symbo and non-arikan kerne, which provide higher minimum distance, and can avoid therefore an error foor. ACKNOWLEDGEMENTS This work was supported by Russian Foundation for Basic Research under the grant a. REFERENCES 1 E. Arikan, Channe poarization: A method for constructing capacityachieving codes for symmetric binary-input memoryess channe, IEEE Transactions On Information Theory, vo. 55, no. 7, pp , Juy S. B. Korada, E. Sasogu, and R. Urbanke, Poar codes: Characterization of exponent, bounds, and constructions, IEEE Transactions On Information Theory, vo. 56, no. 12, pp , December I. Ta and A. Vardy, List decoding of poar codes, in Proceedings of IEEE Internationa Symposium on Information Theory, K. Niu and K. Chen, CRC-aided decoding of poar codes, IEEE Communications Letters, vo. 16, no. 1, October P. Trifonov and V. Miosavskaya, Poar codes with dynamic frozen symbo and their decoding by directed search, in Proceedings of IEEE Information Theory Workshop, September 213, pp V. Miosavskaya and P. Trifonov, Sequentia decoding of poar codes, IEEE Communications Letters, vo. 18, no. 7, pp , U. Wachsmann, R. F. H. Fischer, and J. B. Huber, Mutieve codes: Theoretica concepts and practica design rues, IEEE Transactions On Information Theory, vo. 45, no. 5, pp , Juy A. Abedi and A. Khandani, An anaytica method for approximate performance evauation of binary inear bock codes, IEEE Transactions On Communications, vo. 52, no. 2, February I. Dumer, G. Kabatiansky, and C. Tavernier, Soft-decision ist decoding of Reed-Muer codes with inear compexity, in Proceedings of IEEE Internationa Symposium on Information Theory, A. Vaembois and M. Fossorier, Box and match techniques appied to soft-decision decoding, IEEE Transactions on Information Theory, vo. 5, no. 5, May M. P. Fossorier and S. Lin, Soft-decision decoding of inear bock codes based on ordered statistics, IEEE Transactions on Information Theory, vo. 41, no. 5, pp , September P. A. Martin, D. Tayor, and M. P. Fossorier, Soft-input soft-output ist-based decoding agorithm, IEEE Transactions On Communications, vo. 52, no. 2, February V. Miosavskaya and P. Trifonov, Design of poar codes with arbitrary kerne, in Proceedings of IEEE Information Theory Workshop, 212, pp