1. Applications of Statistical Physics to Coding Theory Nonlinear Dynamics of Iterative Decoding Systems: Analysis and Applications Ljupco Kocarev University of California, San Diego

2. Applications of Statistical Physics to Coding Theory Summary of the presentation Nonlinear Dynamics of the iterative coding systems Nonlinear Codes with Latin Squares (Quasi-groups) Conclusions

3. Applications of Statistical Physics to Coding Theory When: since 2002 Who: Alex Vardy (UCSD) Gian Mario Maggio (ST, Swiss) Zarko Tasev (Kyocera, USA) Frederic Lehmann (INT, Paris, France) Pater Popovski (Tech University, Denmark) Bartolo Scanavino (Poli Torino, Italy) Why: 1.To understand finite-length iterative decoding algorithms using tools from nonlinear systems and chaos theory; and 2.To exploit chaos theory for enhancing existing iterative coding techniques and invent new classes of error-correcting codes. Sponsors: ARO (MURI), STMicroelectronics, University of California (DIMI)

4. Applications of Statistical Physics to Coding Theory Understanding finite-length iterative coding systems Codes on graphs as spin models (1989) Finite length scaling for iterative coding systems (2004) Renormalization group approach for iterative coding systems (2003) Scale-free networks and error-correction code (2004) Nonlinear dynamics of iterative coding systems (2000) As of today asymptotic behavior (as the block-length tends to infinity) of iterative coding systems is reasonably well understood Tom Richardson: The geometry of turbo decoding dynamics (2000)

5. Applications of Statistical Physics to Coding Theory Kalman’s example R. E. Kalman, 1956 “Nonlinear aspects of sampled-data control systems” x n+1 xnxn 1/32/3 1 1/3 10 0 1 S 1 0 S 00 01 Markov process with transition probabilities: 01 1/31/2 2/3 1/2 Logistic map: x n+1 = a x n ( 1 – x n ) a = 3.839

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7. Turbo codes n=1024 Classical turbo codes: C. Berrou, A Glavieux, and P. Thitimajshima, “Near Shannon limit error-correcting coding and decoding: Turbo-Codes,” Proc. IEEE International Communications Conference, pp. 1064-70, 1993

8. Applications of Statistical Physics to Coding Theory Turbo-decoding algorithm BCJR algorithm, 1974 AWGN channel X 1, X 2 - extrinsic information exchanged by the two SISO decoders c 0, c 1, c 2 - channel outputs corresponding to the input sequences

9. Applications of Statistical Physics to Coding Theory The system depends smoothly on its 2n variables X 1, X 2 and 3n parameters c 0, c 1, c 2 (Richardson) E represents a posteriori average entropy Three types of plots: E(l) versus l, E(l+1) versus E(l), and E versus SNR.

10. Applications of Statistical Physics to Coding Theory Bifurcation diagram

11. Applications of Statistical Physics to Coding Theory Discrete-time Hopf bifurcation (part I) -6.7dB-6.5dB -5.9dB -6.3dB-6.1dB 0.75dB0.8dB0.85dB

12. Applications of Statistical Physics to Coding Theory Discrete-time Hopf bifurcation (part II) -6.7dB-6.6dB-6.5dB -6.3dB-6.2dB-6.1dB

13. Applications of Statistical Physics to Coding Theory Example: 2D map Attracting fixed point Attracting invariant curve Chaotic attractor a = 1.9, 2.1, 2.16, 2.27

14. Applications of Statistical Physics to Coding Theory Tangent bifurcation (part I) -7.65dB-7.645dB-7.6dB 0.3dB0.35dB0.4dB

15. Applications of Statistical Physics to Coding Theory Tangent bifurcation (part II) -7.65dB -7.64dB

16. Applications of Statistical Physics to Coding Theory One-dimensional approximation

17. Applications of Statistical Physics to Coding Theory One-dimensional approximation

18. Applications of Statistical Physics to Coding Theory Lyapunov exponents

19. Applications of Statistical Physics to Coding Theory Transient chaos The unequivocal fixed point becomes stable around –1.5dB. Region 0.25dB to 1.25dB (waterfall region): transient chaos Average chaotic transient lifetime for SNR=0.8dB is 378 iteration

20. Applications of Statistical Physics to Coding Theory Transient chaos For each SNR, we generate 1000 different noise realizations (1000 different parameters frames) and compute the number of decoding trajectories that approach the fixed point in less than a given number of iterations. At SNR of 0.6dB, there are 492 frames that converge to the unequivocal fixed point in 5 or less iterations, another 226 frames converge in 10 or less iterations, and so on, while 58 frames remain chaotic after 2000 iterations (which means that their trajectory either approaches a chaotic attractor or that the transient chaos lifetime is very large).

21. Applications of Statistical Physics to Coding Theory Stability of the fixed point

22. Applications of Statistical Physics to Coding Theory LDPC codes Torus-breakdown route to chaos for the regular (216,3,6) LDPC code The values of SNR corresponding to the invariant sets, starting from the fixed point towards chaos, are: 1.19 dB, 1.23 dB, 1.27 dB, 1.33 dB, and 1.44 dB

23. Applications of Statistical Physics to Coding Theory Control of transient chaos Average chaotic transient lifetime for SNR=0.8dB is 9 iterations.

24. Applications of Statistical Physics to Coding Theory Control of transient chaos: results

25. Applications of Statistical Physics to Coding Theory Control of transient chaos: results

26. Applications of Statistical Physics to Coding Theory Conclusions Iterative decoding algorithms can be viewed as high-dimensional dynamical systems parameterized by a large number of parameters. However, although the iterative-decoding algorithm is a high-dimensional dynamical system, it apparently has only a few active variables. The waterfall region for all iterative coding systems exhibits reach dynamical behavior: chaotic attractors, transient chaos, multiple attractors and fractal basin boundaries. Applications: 1.We have proposed a simple adaptive technique to control transient chaos in the turbo decoding algorithm. This results in a ultra-fast convergence and a significant gain in terms of BER performance. 2.We have proposed a novel stopping criterion for turbo codes based on the average entropy of an information block. This is shown to reduce the average number of iterations and to benefit from the use of the adaptive control technique.

27. Applications of Statistical Physics to Coding Theory Open problems: Study how the size of the basin of attraction changes with varying SNR. Find how the value of SNR for which the fixed point at origin becomes a stable point related to the threshold which gives the boundary of the error-free region. Study whether the chaotic behavior of the iterative coding systems comes from the presence of cycles in the graph. Investigate how the topological structure of unstable periodic orbits embedded into the chaotic set is related to the topological structure of the factor graph. Use ergodic theory for dynamical systems to study the properties related to a set of trajectories, not a single trajectory. Find models of low-dimensional dynamical systems averaged over ensembles and/or different noise realizations.

28. Applications of Statistical Physics to Coding Theory Latin Squares (Quasigroups) A Latin square of order N is NxN array of numbers from N-symbol alphabet (say 0, 1, 2, … N-1) in which each row and each column contains each symbol exactly once. Latin squares are also linked to algebraic objects known as quasigroups. A quasigroup is defined in terms of a set, Q, of distinct symbols and a binary operation (called multiplication) involving only the elements of Q. A quasigroup's multiplication table turns out to be a Latin square. D. Gligoroski, S. Markovski (since Sep 2004)

29. Applications of Statistical Physics to Coding Theory Given a quasigroup (Q,*) five new operations, so called parastrophes or adjoint operations, can be derived from the operation *. We need only the following one defined by: (Q,\) is also a quasigroup.

30. Applications of Statistical Physics to Coding Theory Quasigroup string transformations e-transformation d-transformation

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32. Let Q + be the set of all nonempty words (i.e. finite strings) formed by the elements of Q.

33. Applications of Statistical Physics to Coding Theory

34. Nonlinear Codes with Latin Squares (Quasigroups) (Q,*) Q = {0, 1, 2, … N-1} M 1, M 2, …, M r information string; each M i is an element of Q L 1, L 2, …, L m m> r Each L i is either M i or 0 C 1, C 2, …, C m codeword

35. Applications of Statistical Physics to Coding Theory k 1, k 2, …, k n leaders each k i is an element of Q

36. Applications of Statistical Physics to Coding Theory (Q,\) is the left parastrophe of the quasigroup (Q,*)

37. Applications of Statistical Physics to Coding Theory Properties of the Code Nonlinear code “Almost random” code C 1, C 2, ?, C 3, …, C m L 1, L 2, ?, ?, …., ? (synchronous and self - synchronized stream ciphers)

38. Applications of Statistical Physics to Coding Theory Example (Q,*) Q = {0, 1, 2, … 15} Quasigroup of order 16 Each M i is a nibble (4-bit string)

39. Applications of Statistical Physics to Coding Theory (72,288) code with rate ¼ r = 18, m = 72

40. Applications of Statistical Physics to Coding Theory Decoding the code BSC D i is a nibble and D (i) is a sub-block of 4 nibbles (16 bits)

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44. If the set S s contains only one element we say the decoding is successful

45. Applications of Statistical Physics to Coding Theory (72,288) code with rate ¼ r = 18, m = 72 B max = 3

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47. Stability of the decoding algorithm Theorem: If the decoding is successful, then the message is recovered with probability 2 -N(1-R) Conjecture: If then the decoding procedure converges and the cardinality of S s is 1. The relative Gilbert-Varshamov (GV) distance

48. Applications of Statistical Physics to Coding Theory Example Reed – Muller code (6,32) rate 3/16 corrects up to 7 errors T otal number of noisy code-words the codes can correct is 4.5 x 10 6 N = 32 B max = 8 (B 2 = 8) m 1 m 2 0 2 0 2 0 2 0 2 0 2 0 2 m 3 0 2 0 2 0 2 0 2 0 2 0 2 0 2 B max = 5 6885 2 ~ 4.7 10 7 (6,32) rate 3/16

49. Applications of Statistical Physics to Coding Theory Conclusions (open problems) Theoretical framework AWGN channel, soft decoding Concatenated codes: S i is intersection of two sets S i (1) and S i (2)