International Symposium on Information Theory Los Angeles, USA – June, 2020 Successive Cancellation Inactivation Decoding for Modified Reed-Muller and eBCH Codes Mustafa Cemil Coşkun 1,2 Joint work with J. Neu 3 and H. D. Pfister 4 1 German Aerospace Center 2 Technical University of Munich 3 Stanford University 4 Duke University
Outline � Introduction � Preliminaries � Successive Cancellation Inactivation Decoding � Numerical Results � Conclusions
Outline � Introduction � Preliminaries � Successive Cancellation Inactivation Decoding � Numerical Results � Conclusions
Page 1/19 M. C. Coşkun · · Introduction Contributions � Successive cancellation (SC) inactivation decoding is introduced as an efficient implementation of SC list (SCL) decoding over the binary erasure channel (BEC) ◦ E.g., easy to handle more than 150 inactivations, corresponding to an SCL decoder with L ≥ 2 150 , for a (8192 , 4096) code 1E. R. Berlekamp, “The technology of error-correcting codes,” Proc. IEEE, May 1980. 2E. Arıkan, “From sequential decoding to channel polarization and back again,” Shannon Lecture, Jun. 2019.
Page 1/19 M. C. Coşkun · · Introduction Contributions � Successive cancellation (SC) inactivation decoding is introduced as an efficient implementation of SC list (SCL) decoding over the binary erasure channel (BEC) ◦ E.g., easy to handle more than 150 inactivations, corresponding to an SCL decoder with L ≥ 2 150 , for a (8192 , 4096) code � Analysis of the number of inactivations required for the ML decoder 1E. R. Berlekamp, “The technology of error-correcting codes,” Proc. IEEE, May 1980. 2E. Arıkan, “From sequential decoding to channel polarization and back again,” Shannon Lecture, Jun. 2019.
Page 1/19 M. C. Coşkun · · Introduction Contributions � Successive cancellation (SC) inactivation decoding is introduced as an efficient implementation of SC list (SCL) decoding over the binary erasure channel (BEC) ◦ E.g., easy to handle more than 150 inactivations, corresponding to an SCL decoder with L ≥ 2 150 , for a (8192 , 4096) code � Analysis of the number of inactivations required for the ML decoder � Numerical results for various code constructions with dynamic frozen bits, including a modified Reed–Muller (RM) code (denoted d-RM) ◦ d-RM codes perform close to the Berlekamp’s random coding bound 1 ◦ Closely related to polarization-adjusted convolutional (PAC) codes 2 1E. R. Berlekamp, “The technology of error-correcting codes,” Proc. IEEE, May 1980. 2E. Arıkan, “From sequential decoding to channel polarization and back again,” Shannon Lecture, Jun. 2019.
Page 2/19 M. C. Coşkun · · Introduction Motivation � The BEC, while a special case, can be useful for the design of good codes for general channels 3 ◦ A good code with small number of inactivations on the BEC may also be good under SCL decoding with small list size on the AWGN channel 3F. Peng, W. Ryan, R. Wesel, “Surrogate-channel design of universal LDPC codes,” CL, Jun. 2006.
Page 2/19 M. C. Coşkun · · Introduction Motivation � The BEC, while a special case, can be useful for the design of good codes for general channels 3 ◦ A good code with small number of inactivations on the BEC may also be good under SCL decoding with small list size on the AWGN channel � The number of inactivations quantifies the average complexity for an ML decoder, implemented via SC inactivation (or SCL) decoding 3F. Peng, W. Ryan, R. Wesel, “Surrogate-channel design of universal LDPC codes,” CL, Jun. 2006.
Outline � Introduction � Preliminaries � Successive Cancellation Inactivation Decoding � Numerical Results � Conclusions
Page 3/19 M. C. Coşkun · · Preliminaries Polar Codes and Density Evolution 4 � � 1 0 1 K ⊗ m x n 1 = u n K 2 � n = 2 m where and 2 1 1 4E. Arıkan, “Channel polarization: A method for constructing capacity-achieving codes for symmetric binary-input memoryless channels,” T-IT, Jul. 2009.
Page 3/19 M. C. Coşkun · · Preliminaries Polar Codes and Density Evolution 4 � � 1 0 1 K ⊗ m x n 1 = u n K 2 � n = 2 m where and 2 1 1 u 1 BEC( 0 . 5 ) y 1 ⊕ ⊕ ⊕ x 1 u 2 BEC( 0 . 5 ) y 2 ⊕ ⊕ • x 2 u 3 BEC( 0 . 5 ) y 3 ⊕ ⊕ • x 3 y 4 u 4 BEC( 0 . 5 ) ⊕ • • x 4 y 5 u 5 ⊕ ⊕ BEC( 0 . 5 ) • x 5 u 6 y 6 ⊕ BEC( 0 . 5 ) • • x 6 u 7 BEC( 0 . 5 ) y 7 ⊕ • • x 7 u 8 BEC( 0 . 5 ) y 8 • • • x 8 4E. Arıkan, “Channel polarization: A method for constructing capacity-achieving codes for symmetric binary-input memoryless channels,” T-IT, Jul. 2009.
Page 3/19 M. C. Coşkun · · Preliminaries Polar Codes and Density Evolution 4 � � 1 0 1 K ⊗ m x n 1 = u n K 2 � n = 2 m where and 2 1 1 F = { 1 , 2 , 3 , 5 } BEC( 0 . 5 ) y 1 frozen ǫ 1 = 0 . 9961 ⊕ ⊕ ⊕ x 1 A = { 4 , 6 , 7 , 8 } ǫ 2 = 0 . 8789 BEC( 0 . 5 ) y 2 frozen ⊕ ⊕ • x 2 ǫ 3 = 0 . 8086 BEC( 0 . 5 ) y 3 frozen ⊕ ⊕ • x 3 y 4 ǫ 4 = 0 . 3164 BEC( 0 . 5 ) info ⊕ • • x 4 y 5 frozen ǫ 5 = 0 . 6836 ⊕ ⊕ BEC( 0 . 5 ) • x 5 y 6 info ǫ 6 = 0 . 1914 ⊕ BEC( 0 . 5 ) • • x 6 BEC( 0 . 5 ) y 7 info ǫ 7 = 0 . 1211 ⊕ • • x 7 ǫ 8 = 0 . 0039 BEC( 0 . 5 ) y 8 info • • • x 8 4E. Arıkan, “Channel polarization: A method for constructing capacity-achieving codes for symmetric binary-input memoryless channels,” T-IT, Jul. 2009.
Page 4/19 M. C. Coşkun · · Preliminaries Dynamic Frozen Bits � The value of a frozen bit can also be set to a linear combination of previous information bits (rather than 0 ) � A frozen bit whose value depends on past inputs is called dynamic � The SC/SCL decoders are easily modified to decode polar codes with dynamic frozen bits 5 � Any binary linear block code can be represented as a polar code with dynamic frozen bits 5 (e.g., PAC code) 5 P. Trifonov, V. Miloslavskaya, “Polar subcodes,” J-SAC, Feb. 2016.
Outline � Introduction � Preliminaries � Successive Cancellation Inactivation Decoding � Numerical Results � Conclusions
Page 5/19 M. C. Coşkun · · Successive Cancellation Inactivation Decoding Linear Codes over Erasure Channels � For linear codes on the BEC, any uncertainty in the information bits always takes the form of an affine subspace 6T. J. Richardson, R. L. Urbanke, “Efficient encoding of low-density parity-check codes,” T-IT, Feb. 2001. 7H. Pishro-Nik, F. Fekri, “On decoding of low-density parity-check codes over the binary erasure channel,” T-IT, Mar. 2004. 8C. Méasson, A. Montanari, R. Urbanke, “Maxwell construction: The hidden bridge between iterative and maximum a posteriori decoding,” T-IT, Dec. 2008. 9A. Shokrollahi, “Raptor codes,” T-IT, Dec. 2008. 10A. Eslami, H. Pishro-Nik, “On bit error rate performance of polar codes in finite regime,” 48th Annu. Allerton Conf. on Commun., Control, and Comput., Sep. 2010.
Page 5/19 M. C. Coşkun · · Successive Cancellation Inactivation Decoding Linear Codes over Erasure Channels � For linear codes on the BEC, any uncertainty in the information bits always takes the form of an affine subspace � The SCL decoder on the BEC lists all valid paths in this subspace � The SC inactivation decoder stores a basis instead of listing all valid paths 6T. J. Richardson, R. L. Urbanke, “Efficient encoding of low-density parity-check codes,” T-IT, Feb. 2001. 7H. Pishro-Nik, F. Fekri, “On decoding of low-density parity-check codes over the binary erasure channel,” T-IT, Mar. 2004. 8C. Méasson, A. Montanari, R. Urbanke, “Maxwell construction: The hidden bridge between iterative and maximum a posteriori decoding,” T-IT, Dec. 2008. 9A. Shokrollahi, “Raptor codes,” T-IT, Dec. 2008. 10A. Eslami, H. Pishro-Nik, “On bit error rate performance of polar codes in finite regime,” 48th Annu. Allerton Conf. on Commun., Control, and Comput., Sep. 2010.
Page 5/19 M. C. Coşkun · · Successive Cancellation Inactivation Decoding Linear Codes over Erasure Channels � For linear codes on the BEC, any uncertainty in the information bits always takes the form of an affine subspace � The SCL decoder on the BEC lists all valid paths in this subspace � The SC inactivation decoder stores a basis instead of listing all valid paths � Similar methods have been proposed for low-density parity-check 6 7 8 and raptor 9 codes � For polar codes, a BP decoder with inactivations was proposed, 10 improving bit-error rate, but it does not use SC decoding schedule 6T. J. Richardson, R. L. Urbanke, “Efficient encoding of low-density parity-check codes,” T-IT, Feb. 2001. 7H. Pishro-Nik, F. Fekri, “On decoding of low-density parity-check codes over the binary erasure channel,” T-IT, Mar. 2004. 8C. Méasson, A. Montanari, R. Urbanke, “Maxwell construction: The hidden bridge between iterative and maximum a posteriori decoding,” T-IT, Dec. 2008. 9A. Shokrollahi, “Raptor codes,” T-IT, Dec. 2008. 10A. Eslami, H. Pishro-Nik, “On bit error rate performance of polar codes in finite regime,” 48th Annu. Allerton Conf. on Commun., Control, and Comput., Sep. 2010.
Page 6/19 M. C. Coşkun · · Successive Cancellation Inactivation Decoding The Algorithm � The SC inactivation decoder has the same message passing schedule as the SC decoder � Whenever an information bit is decoded as erased, it is replaced by a dummy variable (i.e., inactivated)
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