Successive Cancellation Decoding of Single Parity-Check Product - PowerPoint PPT Presentation

Institute for Communications Engineering Technische Universit¨ at M¨ unchen Successive Cancellation Decoding of Single Parity-Check Product Codes: Analysis and Improved Decoding Algorithms skun 1 , 2 Mustafa Cemil Co¸ Joint work with Gianluigi Liva 2 , Alexandre Graell i Amat 3 , Michael Lentmaier 4 and Thomas Jerkovits 2 1 Institute for Communications Engineering, Technical University of Munich, Munich, Germany 2 Institute for Communication and Navigation, German Aerospace Center, Weßling, Germany 3 Department of Signals and Systems, Chalmers University of Technology, Gothenburg, Sweden 4 Department of Electrical and Information Technology, Lund University, Lund, Sweden LAWCI, Campinas - July, 2018

Institute for Communications Engineering Technische Universit¨ at M¨ unchen Basics on Product Codes 1 The first deterministic code construction allowing vanishing bit error probability at a non-zero rate Very popular: IEEE 802.16 (Wimax) ITU-G975 (Optical) Storage 1 P. Elias, “Error-free coding,” IRE T-IT , Sep. 1954. Mustafa Cemil Co¸ skun — SC Decoding of Single Parity-Check Product Codes: Analysis and Improved Decoding Algorithms 1/5

Institute for Communications Engineering Technische Universit¨ at M¨ unchen Basics on Product Codes u 0 u 1 u 2 u 3 u 4 u 5 u 6 u 7 u 8 u 9 u 10 u 11 Mustafa Cemil Co¸ skun — SC Decoding of Single Parity-Check Product Codes: Analysis and Improved Decoding Algorithms 2/5

Institute for Communications Engineering Technische Universit¨ at M¨ unchen Basics on Product Codes Row Encoding: ( n 1 , k 1 ) code C 1 u 0 u 1 u 2 u 3 p 0 p 1 p 2 u 4 u 5 u 6 u 7 u 8 u 9 u 10 u 11 Mustafa Cemil Co¸ skun — SC Decoding of Single Parity-Check Product Codes: Analysis and Improved Decoding Algorithms 2/5

Institute for Communications Engineering Technische Universit¨ at M¨ unchen Basics on Product Codes Row Encoding: ( n 1 , k 1 ) code C 1 u 0 u 1 u 2 u 3 p 0 p 1 p 2 u 4 u 5 u 6 u 7 p 3 p 4 p 5 u 8 u 9 u 10 u 11 Mustafa Cemil Co¸ skun — SC Decoding of Single Parity-Check Product Codes: Analysis and Improved Decoding Algorithms 2/5

Institute for Communications Engineering Technische Universit¨ at M¨ unchen Basics on Product Codes Row Encoding: ( n 1 , k 1 ) code C 1 u 0 u 1 u 2 u 3 p 0 p 1 p 2 u 4 u 5 u 6 u 7 p 3 p 4 p 5 u 8 u 9 u 10 u 11 p 6 p 7 p 8 Mustafa Cemil Co¸ skun — SC Decoding of Single Parity-Check Product Codes: Analysis and Improved Decoding Algorithms 2/5

Institute for Communications Engineering Technische Universit¨ at M¨ unchen Basics on Product Codes Row Encoding: ( n 1 , k 1 ) code C 1 u 0 u 1 u 2 u 3 p 0 p 1 p 2 u 4 u 5 u 6 u 7 p 3 p 4 p 5 u 8 u 9 u 10 u 11 p 6 p 7 p 8 Column Encoding p 9 ( n 2 , k 2 ) code C 2 p 10 p 11 p 12 Mustafa Cemil Co¸ skun — SC Decoding of Single Parity-Check Product Codes: Analysis and Improved Decoding Algorithms 2/5

Institute for Communications Engineering Technische Universit¨ at M¨ unchen Basics on Product Codes Row Encoding: ( n 1 , k 1 ) code C 1 u 0 u 1 u 2 u 3 p 0 p 1 p 2 u 4 u 5 u 6 u 7 p 3 p 4 p 5 u 8 u 9 u 10 u 11 p 6 p 7 p 8 Column Encoding p 9 p 13 ( n 2 , k 2 ) code C 2 p 10 p 14 p 11 p 15 p 12 p 16 Mustafa Cemil Co¸ skun — SC Decoding of Single Parity-Check Product Codes: Analysis and Improved Decoding Algorithms 2/5

Institute for Communications Engineering Technische Universit¨ at M¨ unchen Basics on Product Codes Row Encoding: ( n 1 , k 1 ) code C 1 u 0 u 1 u 2 u 3 p 0 p 1 p 2 u 4 u 5 u 6 u 7 p 3 p 4 p 5 u 8 u 9 u 10 u 11 p 6 p 7 p 8 Column Encoding p 9 p 13 p 17 ( n 2 , k 2 ) code C 2 p 10 p 14 p 18 p 11 p 15 p 19 p 12 p 16 p 20 Mustafa Cemil Co¸ skun — SC Decoding of Single Parity-Check Product Codes: Analysis and Improved Decoding Algorithms 2/5

Institute for Communications Engineering Technische Universit¨ at M¨ unchen Basics on Product Codes Row Encoding: ( n 1 , k 1 ) code C 1 u 0 u 1 u 2 u 3 p 0 p 1 p 2 u 4 u 5 u 6 u 7 p 3 p 4 p 5 u 8 u 9 u 10 u 11 p 6 p 7 p 8 Column Encoding p 9 p 13 p 17 p 21 ( n 2 , k 2 ) code C 2 p 10 p 14 p 18 p 22 p 11 p 15 p 19 p 23 p 12 p 16 p 20 p 24 Mustafa Cemil Co¸ skun — SC Decoding of Single Parity-Check Product Codes: Analysis and Improved Decoding Algorithms 2/5

Institute for Communications Engineering Technische Universit¨ at M¨ unchen Basics on Product Codes Row Encoding: ( n 1 , k 1 ) code C 1 u 0 u 1 u 2 u 3 p 0 p 1 p 2 u 4 u 5 u 6 u 7 p 3 p 4 p 5 u 8 u 9 u 10 u 11 p 6 p 7 p 8 Column Encoding p 9 p 13 p 17 p 21 p 25 ( n 2 , k 2 ) code C 2 p 10 p 14 p 18 p 22 p 26 p 11 p 15 p 19 p 23 p 27 p 12 p 16 p 20 p 24 p 28 Mustafa Cemil Co¸ skun — SC Decoding of Single Parity-Check Product Codes: Analysis and Improved Decoding Algorithms 2/5

Institute for Communications Engineering Technische Universit¨ at M¨ unchen Basics on Product Codes Row Encoding: ( n 1 , k 1 ) code C 1 u 0 u 1 u 2 u 3 p 0 p 1 p 2 u 4 u 5 u 6 u 7 p 3 p 4 p 5 u 8 u 9 u 10 u 11 p 6 p 7 p 8 Column Encoding p 9 p 13 p 17 p 21 p 25 p 29 ( n 2 , k 2 ) code C 2 p 10 p 14 p 18 p 22 p 26 p 30 p 11 p 15 p 19 p 23 p 27 p 31 p 12 p 16 p 20 p 24 p 28 p 32 Mustafa Cemil Co¸ skun — SC Decoding of Single Parity-Check Product Codes: Analysis and Improved Decoding Algorithms 2/5

Institute for Communications Engineering Technische Universit¨ at M¨ unchen Basics on Product Codes Row Encoding: ( n 1 , k 1 ) code C 1 u 0 u 1 u 2 u 3 p 0 p 1 p 2 u 4 u 5 u 6 u 7 p 3 p 4 p 5 u 8 u 9 u 10 u 11 p 6 p 7 p 8 Column Encoding p 9 p 13 p 17 p 21 p 25 p 29 p 33 ( n 2 , k 2 ) code C 2 p 10 p 14 p 18 p 22 p 26 p 30 p 34 p 11 p 15 p 19 p 23 p 27 p 31 p 35 p 12 p 16 p 20 p 24 p 28 p 32 p 36 Mustafa Cemil Co¸ skun — SC Decoding of Single Parity-Check Product Codes: Analysis and Improved Decoding Algorithms 2/5

Institute for Communications Engineering Technische Universit¨ at M¨ unchen Basics on Product Codes Row Encoding: ( n 1 , k 1 ) code C 1 u 0 u 1 u 2 u 3 p 0 p 1 p 2 u 4 u 5 u 6 u 7 p 3 p 4 p 5 u 8 u 9 u 10 u 11 p 6 p 7 p 8 Column Encoding p 9 p 13 p 17 p 21 p 25 p 29 p 33 ( n 2 , k 2 ) code C 2 p 10 p 14 p 18 p 22 p 26 p 30 p 34 p 11 p 15 p 19 p 23 p 27 p 31 p 35 p 12 p 16 p 20 p 24 p 28 p 32 p 36 Mustafa Cemil Co¸ skun — SC Decoding of Single Parity-Check Product Codes: Analysis and Improved Decoding Algorithms 2/5

Institute for Communications Engineering Technische Universit¨ at M¨ unchen Last year in ISIT 2 ... Applied successive cancellation decoding (SCD) to product codes with single parity check component codes (SPC-PCs) Theoretical bounds on P B under both SCD and Elias’ decoding over the binary erasure channel (BEC) Showed that P B under SCD is upper bounded by the one under Elias’ decoding over the BEC 2 M. C. Coskun, G. Liva, A. G. i Amat, M. Lentmaier, “Successive Cancellation Decoding of Single Parity-Check Product Codes,” IEEE ISIT , June 2017. Mustafa Cemil Co¸ skun — SC Decoding of Single Parity-Check Product Codes: Analysis and Improved Decoding Algorithms 3/5

Institute for Communications Engineering Technische Universit¨ at M¨ unchen As an extension... Provide some asymptotic constructions with a positive rate and vanishing error probability over the BEC Show that they can be seen as a ”polar code” with larger kernels and different selection of frozen bits Introduce successive cancellation list decoding+outer CRC 3 Comments on open problems... 3 I. Tal and A. Vardy, “List decoding of polar codes,” Proc. ISIT , 2011. Mustafa Cemil Co¸ skun — SC Decoding of Single Parity-Check Product Codes: Analysis and Improved Decoding Algorithms 4/5

Institute for Communications Engineering Technische Universit¨ at M¨ unchen THANKS! Mustafa Cemil Co¸ skun — SC Decoding of Single Parity-Check Product Codes: Analysis and Improved Decoding Algorithms 5/5