Exponential Error Bounds for Block Concatenated Codes with Tail Biting Trellis Inner Codes

1. 2004 International Symposium of Information Theory and its Applications Oct. 10-13, 2004, Parma, ITALY Exponential Error Bounds for Block Concatenated Codes with Tail Biting Trellis Inner Codes Masao Kasahara Faculty of Informatics, Osaka Gakuin University, Osaka, JAPAN kasahara@utc.osaka-gu.ac.jp Shigeich Hirasawa Department of Industrial and Management Systems Engineering School of Science and EngineeringWaseda University, Tokyo, JAPAN hirasawa@hirasa.mgmt.waseda.ac.jp

2. 1. Introduction 1. Introduction Coding Theorem … random coding arguments • existence of a code • essential behavior of the code • quantitative evaluation : probability of decoding error : rate : decoding complexity Coding theorem aspects → practical coding problem

3. 1. Introduction • Tail biting trellis codes c.f. Ordinary block codes • Block concatenated codes with tail biting trellis inner codes --- Codes • inner code --- tail biting trellis code • outer code --- Reed Solomon (RS) code • c.f. Ordinary block concatenated codes proposed by G.D. Forney, Jr. [1] • inner code --- ordinary block code • outer code --- RS code • Parallel concatenated codes • Serial concatenated codes

4. 2. Preliminaries 2. Preliminaries

5. 2. Preliminaries

6. 2. Preliminaries Coding theorem Block Code : Trellis code :

7. 2. Preliminaries e(r) E(R) Fig.0: Block code exponent and trellis code exponent for very noisy channel

8. 2. Preliminaries

9. 2. Preliminaries

10. 2. Preliminaries

11. 3. Tail biting trellis codes 3. Tail biting trellis codes Trellis diagram states trellis diagrams

12. 3. Tail biting trellis codes : true path incorrected path

13. 3. Tail biting trellis codes true path :

14. 3. Tail biting trellis codes

15. 3. Tail biting trellis codes

16. 3. Tail biting trellis codes

17. 3. Tail biting trellis codes

18. 3. Tail biting trellis codes

19. 3. Tail biting trellis codes Fig.1: Negative exponents in terms of G for very noisy channel

20. 4. Concatenated codes with tail biting trellis inner codes 4. Concatenated codes with Tail biting trellis inner codes

21. 4. Concatenated codes with Tail biting trellis inner codes

22. 4. Concatenated codes with Tail biting trellis inner codes

23. 4. Concatenated codes with Tail biting trellis inner codes

24. 4. Concatenated codes with Tail biting trellis inner codes

26. 4. Concatenated codes with Tail biting trellis inner codes Fig.2: Error exponents for code and code for very noisy channel

27. 5. Generalized version of concatenated codes with tail biting trellis codes

28. 6. Concluding remarks (1) (2) (3)