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## 2 Recursive Descriptions of Decoding Algorithms and Hardware Architectures for Polar Codes

### Citations

254 |
Polarization: A Method for Constructing Capacity-achieving Codes for Symmetric Binary-Input Memoryless Channels
- Arıkan, “Channel
- 2009
(Show Context)
Citation Context ... for these decoding algorithms are also described in a recursive way, both for Arikan’s standard polar codes and for arbitrary polarizing kernels. 1 Introduction Polar codes were introduced by Arikan =-=[1]-=- and provided a scheme for achieving the symmetric capacity of binary memoryless channels (B-MC) with polynomial encoding and decoding complexities. Arikan used the so-called (u+ v, v) construction, w... |

239 |
Concatenated Codes
- Forney
- 1966
(Show Context)
Citation Context ...ar codes. For example, if we take the first kernel g1(u0, u1,2, u3) = x 3 0 ∈ {0, 1} 4 and define the RS kernel 1The construction of the GCCs is a generalization of Forney’s code concatenation method =-=[17]-=-. 4 Figure 3: Mixed Kernel as GCC as g2(u0,1, u2,3, u4,5, u6,7) = x 3 0 ∈ ( {0, 1}2 )4 [7], then the general concatenated construction is given in Figure 3. Now, note that using g (m) 2 mapping over a... |

178 |
Codes on graphs: Normal realizations
- Forney
- 2001
(Show Context)
Citation Context ...p. 3.3 A Recursive Description of the BP Algorithm BP is an alternative to SC decoding [1]. It is an iterative message-passing algorithm, which messages are defined using Forney’s normal factor graph =-=[23]-=-. There is no evidence which algorithm is better for general channels, except for the BEC, in which BP is shown to outperform SC [12]. However, simulations indicate that BP outperforms SC in many case... |

87 | On the rate of channel polarization,” in
- Arıkan, Telatar
- 2009
(Show Context)
Citation Context ...Arikan showed that the SC decoding algorithm has an algorithmic time and space complexity which is O(N · log(N)) (The same complexity holds also for the encoding algorithm). Furthermore, it was shown =-=[2]-=- that asymptotically in the block length N , the block error probability of this scheme decays to zero like O(2− √ N ). Generalizations of Arikan’s code structures were soon to follow. Korada et al. c... |

72 | Polar codes for channel and source coding”,
- Korada
- 2009
(Show Context)
Citation Context ...orithm . This is a message passing iterative decoding algorithm that operates on the normal factor graph representation of the code. It is known to outperform SC over the Binary Erasure Channel (BEC) =-=[12]-=-, and seems to have good performance on other channels as well [12, 13]. Leroux et al. considered efficient hardware implementations for the SC decoder for the (u+ v, v) polar code [14, 15]. They gave... |

46 |
List decoding of polar codes,”
- Tal, Vardy
- 2011
(Show Context)
Citation Context ...ence, and we elaborate on it in the sequel. Generalizations and alternatives to SC as the decoding algorithm were also studied. Tal and Vardy introduced the Successive Cancellation List (SCL) decoder =-=[9, 10]-=-. In this algorithm, the decoder consider up to M concurrent decoding paths at each one of its stages, where M is the size of the list. At the final stage of the algorithm, the most likely result is s... |

42 | Polar codes: characterization of exponent, bounds, and constructions,”
- Korada, oglu, et al.
- 2010
(Show Context)
Citation Context ...ngth N , the block error probability of this scheme decays to zero like O(2− √ N ). Generalizations of Arikan’s code structures were soon to follow. Korada et al. considered binary and linear kernels =-=[3]-=-. They showed that a binary linear kernel is polarizing if and only if its corresponding generating matrix is upper-triangular, and analyzed their rate of polarization, by introducing the notion of ke... |

23 | Channel polarization on q-ary discrete memoryless channels by arbitrary kernels
- Mori, Tanaka
(Show Context)
Citation Context ...introducing the notion of kernel exponent. Mori and Tanaka considered the general case of a mapping g(·), which is not necessarily linear and binary, as a basis for channel polarization constructions =-=[4]-=-. They gave sufficient conditions for polarization and generalized the exponent for these cases. They further showed examples of linear and non-binary Reed-Solomon codes and Algebraic Geometry with ex... |

21 | Hardware architectures for successive cancellation decoding of polar codes
- Leroux, Tal, et al.
- 2011
(Show Context)
Citation Context ... Channel (BEC) [12], and seems to have good performance on other channels as well [12, 13]. Leroux et al. considered efficient hardware implementations for the SC decoder for the (u+ v, v) polar code =-=[14, 15]-=-. They gave an explicit design of a ”line decoder” with N/2 processing elements and O(N) memory elements. Their work, contains an efficient approximate min-sum decoder, and a discussion on a fixed poi... |

14 | Efficient serial message-passing schedules for LDPC decoding
- Sharon, Litsyn, et al.
- 2007
(Show Context)
Citation Context ...se, one can stop an iteration in the middle by holding a counter in a similar way to the method that is usually used in BP decoding of LDPC codes using the check-node based serial schedules (see e.g. =-=[24]-=-). We note, however, that in the LDPC case, the consistency is manifested in the fact that all the parity check equations are satisfied. In the next section we describe hardware architectures for the ... |

8 |
An Adaptive Successive Cancellation List Decoder for Polar Codes with Cyclic Redundancy
- Li, Shen, et al.
- 2012
(Show Context)
Citation Context ...ize for each outer code, in the GCC structure of the polar code. This approach seems to give better results, comparing it to standard list approach with the same average list size. Finally, Li et al. =-=[11]-=-, suggested an iterative SCL with CRC algorithm in which the decoder increases the list size by a multiplicative factor of 2 and restart the algorithm, if at the end of the SCL algorithm there doesn’t... |

6 |
Coding of generalized concatenated codes,” Probl
- Blokh, Zyablov
- 1974
(Show Context)
Citation Context ...+1)·ℓ i·ℓm−1+1 ) 0 ≤ i ≤ ℓm−1 − 1 0 ≤ j ≤ ℓ− 1. General Concatenated Codes (GCC)1 are error correcting codes that are generated by a construction technique, which was introduced by Blokh and Zyabolov =-=[18]-=- and Zinoviev [19]. In this construction, we have ℓ outer codes {Cr} ℓ−1 r=0, where Cr is an Nout length code of size Mr over alphabet Fr. We also have an inner code of length Nin and size ∏ℓ−1 r=0 |F... |

6 |
Soft-decision decoding of reed-muller codes: recursive lists
- Dumer, Shabunov
- 2006
(Show Context)
Citation Context ...s in Dumer’s survey on GCCs [20]. In addition, the recursive decoding algorithms for Reed-Muller (RM) codes, utilizing their Plotkin (u+v, v) recursive GCC structure were extensively studied by Dumer =-=[21, 22]-=- and are closely related to the algorithms we present here. Actually, Dumer’s simplified decoding algorithm for RM codes [22, Section IV] is the SC decoding for the Arikan’s structure, we describe in ... |

5 | An FPGA implementation architecture for decoding of polar codes - Pamuk - 2011 |

3 |
polar codes using Reed-Solomon codes and algebraic geometry codes
- “Non-binary
(Show Context)
Citation Context ...nent for these cases. They further showed examples of linear and non-binary Reed-Solomon codes and Algebraic Geometry with exponents that are far better than the exponents of the known binary kernels =-=[5]-=-. The authors of this correspondence gave examples of binary but non-linear kernels having the optimal exponent per their kernel dimensions [6]. All of these structures ∗Noam Presman and Simon Litsyn ... |

3 |
Hardware implementation of successive-cancellation decoders for polar codes,”
- Leroux, Raymond, et al.
- 2012
(Show Context)
Citation Context ... Channel (BEC) [12], and seems to have good performance on other channels as well [12, 13]. Leroux et al. considered efficient hardware implementations for the SC decoder for the (u+ v, v) polar code =-=[14, 15]-=-. They gave an explicit design of a ”line decoder” with N/2 processing elements and O(N) memory elements. Their work, contains an efficient approximate min-sum decoder, and a discussion on a fixed poi... |

3 | Soft-decision decoding of reed-muller codes: a simplified algorithm
- Dumer
- 2006
(Show Context)
Citation Context ...s in Dumer’s survey on GCCs [20]. In addition, the recursive decoding algorithms for Reed-Muller (RM) codes, utilizing their Plotkin (u+v, v) recursive GCC structure were extensively studied by Dumer =-=[21, 22]-=- and are closely related to the algorithms we present here. Actually, Dumer’s simplified decoding algorithm for RM codes [22, Section IV] is the SC decoding for the Arikan’s structure, we describe in ... |

2 | Binary polar code kernels from code decompositions
- Presman, Shapira, et al.
- 2011
(Show Context)
Citation Context ...ar better than the exponents of the known binary kernels [5]. The authors of this correspondence gave examples of binary but non-linear kernels having the optimal exponent per their kernel dimensions =-=[6]-=-. All of these structures ∗Noam Presman and Simon Litsyn are with the the School of Electrical Engineering, Tel Aviv University, Ramat Aviv 69978 Israel. (e-mails: {presmann, litsyn}@eng.tau.ac.il.). ... |

2 |
decoding of polar codes
- “List
- 2012
(Show Context)
Citation Context ...ence, and we elaborate on it in the sequel. Generalizations and alternatives to SC as the decoding algorithm were also studied. Tal and Vardy introduced the Successive Cancellation List (SCL) decoder =-=[9, 10]-=-. In this algorithm, the decoder consider up to M concurrent decoding paths at each one of its stages, where M is the size of the list. At the final stage of the algorithm, the most likely result is s... |

1 |
Efficient design and decoding of polar codes.” [Online]. Available: http://dcn.infos.ru/∼petert
- Trifonov
(Show Context)
Citation Context ... in the so-called mixed kernel structure, that have demonstrated good performance for finite length codes in many cases. A further generalization of the polar code structure was suggested by Trifonov =-=[8]-=-, in which the outer polar codes were replaced by suitable codes along with their appropriate decoding algorithms. We note here, that the representation of polar codes as instances of general concaten... |

1 |
A performance comparison of polar codes and reed-muller codes
- Arkan
- 2008
(Show Context)
Citation Context ...t operates on the normal factor graph representation of the code. It is known to outperform SC over the Binary Erasure Channel (BEC) [12], and seems to have good performance on other channels as well =-=[12, 13]-=-. Leroux et al. considered efficient hardware implementations for the SC decoder for the (u+ v, v) polar code [14, 15]. They gave an explicit design of a ”line decoder” with N/2 processing elements an... |

1 |
Generalized concatenated codes,” Probl
- Zinoviev
- 1976
(Show Context)
Citation Context ... ≤ i ≤ ℓm−1 − 1 0 ≤ j ≤ ℓ− 1. General Concatenated Codes (GCC)1 are error correcting codes that are generated by a construction technique, which was introduced by Blokh and Zyabolov [18] and Zinoviev =-=[19]-=-. In this construction, we have ℓ outer codes {Cr} ℓ−1 r=0, where Cr is an Nout length code of size Mr over alphabet Fr. We also have an inner code of length Nin and size ∏ℓ−1 r=0 |Fr| over alphabet F... |

1 |
Handbook of Coding Theory
- Dumer
- 1998
(Show Context)
Citation Context .... It is created by taking an ℓ × Nout matrix, in which the r th row is a codeword from Cr, and applying the inner mapping φ on each of the Nout columns of the matrix. As Dumer describes in his survey =-=[20]-=-, the GCCs can give good code parameters for short length codes due to a good combination of outer codes and a nested inner code. In fact, some of them give the best parameters known. Moreover, it is ... |

1 |
Design of binary polar codes with arbitrary kernel
- Miloslavskaya, Trifonov
(Show Context)
Citation Context ...onsider trellis implementation, of the decoding stages, or even consider using approximations of it, such as min-sum rule [15], or near ML decoding variants, such as order statistics or box and match =-=[25]-=-. Since the outer codes in this design are of length N/ℓ, the processors in the preparatory steps of the SC algorithm (i.e. steps 2 · r − 1, as defined in Section 3) should generate N/ℓ llr functions,... |