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List-Decoding of Polar Codes Ido Tal and Alexander Vardy University - PowerPoint PPT Presentation

Nov 20, 2023 •454 likes •1.53k views

List-Decoding of Polar Codes Ido Tal and Alexander Vardy University of California San Diego 9500 Gilman Drive, La Jolla, CA 92093, USA Problem and goal Channel polarization is slow. For short to moderate code lengths, polar codes have

  1. A closer look at successive cancellation ( P ( y 0 y 1 | u 0 = 0 ) , P ( y 0 y 1 | u 0 = 1 )) ( P ( y 0 | x 0 = 0 ) , P ( y 0 | x 0 = 1 )) x 0 → y 0 u 0 u 0 � x 0 � ( P ( y 0 y 1 � u 0 | u 1 = 0 ) , P ( y 0 y 1 � u 0 | u 1 = 1 )) ( P ( y 1 | x 1 = 0 ) , P ( y 1 | x 1 = 1 )) x 1 → y 1 u 1 � u 1 x 1 � probability pair variable boolean variable (bit)

  2. A closer look at successive cancellation ? ? ( P ( y 0 y 1 | u 0 = 0 ) , P ( y 0 y 1 | u 0 = 1 )) ( P ( y 0 | x 0 = 0 ) , P ( y 0 | x 0 = 1 )) x 0 → y 0 u 0 u 0 � ? ? x 0 � ? ? ( P ( y 0 y 1 � u 0 | u 1 = 0 ) , P ( y 0 y 1 � u 0 | u 1 = 1 )) ( P ( y 1 | x 1 = 0 ) , P ( y 1 | x 1 = 1 )) x 1 → y 1 u 1 � u 1 ? ? x 1 � probability pair variable boolean variable (bit)

  3. A closer look at successive cancellation ? � ( P ( y 0 y 1 | u 0 = 0 ) , P ( y 0 y 1 | u 0 = 1 )) ( P ( y 0 | x 0 = 0 ) , P ( y 0 | x 0 = 1 )) x 0 → y 0 u 0 u 0 � ? ? x 0 � ? � ( P ( y 0 y 1 � u 0 | u 1 = 0 ) , P ( y 0 y 1 � u 0 | u 1 = 1 )) ( P ( y 1 | x 1 = 0 ) , P ( y 1 | x 1 = 1 )) x 1 → y 1 u 1 � u 1 ? ? x 1 � probability pair variable boolean variable (bit)

  4. A closer look at successive cancellation � � ( P ( y 0 y 1 | u 0 = 0 ) , P ( y 0 y 1 | u 0 = 1 )) ( P ( y 0 | x 0 = 0 ) , P ( y 0 | x 0 = 1 )) x 0 → y 0 u 0 u 0 � ? ? x 0 � ? � ( P ( y 0 y 1 � u 0 | u 1 = 0 ) , P ( y 0 y 1 � u 0 | u 1 = 1 )) ( P ( y 1 | x 1 = 0 ) , P ( y 1 | x 1 = 1 )) x 1 → y 1 u 1 � u 1 ? ? x 1 � probability pair variable boolean variable (bit)

  5. A closer look at successive cancellation � � ( P ( y 0 y 1 | u 0 = 0 ) , P ( y 0 y 1 | u 0 = 1 )) ( P ( y 0 | x 0 = 0 ) , P ( y 0 | x 0 = 1 )) x 0 → y 0 u 0 u 0 � � ? x 0 � ? � ( P ( y 0 y 1 � u 0 | u 1 = 0 ) , P ( y 0 y 1 � u 0 | u 1 = 1 )) ( P ( y 1 | x 1 = 0 ) , P ( y 1 | x 1 = 1 )) x 1 → y 1 u 1 � u 1 ? ? x 1 � probability pair variable boolean variable (bit)

  6. A closer look at successive cancellation � � ( P ( y 0 y 1 | u 0 = 0 ) , P ( y 0 y 1 | u 0 = 1 )) ( P ( y 0 | x 0 = 0 ) , P ( y 0 | x 0 = 1 )) x 0 → y 0 u 0 u 0 � � ? x 0 � ? � ( P ( y 0 y 1 � u 0 | u 1 = 0 ) , P ( y 0 y 1 � u 0 | u 1 = 1 )) ( P ( y 1 | x 1 = 0 ) , P ( y 1 | x 1 = 1 )) x 1 → y 1 u 1 � u 1 ? ? x 1 � probability pair variable boolean variable (bit)

  7. A closer look at successive cancellation � � ( P ( y 0 y 1 | u 0 = 0 ) , P ( y 0 y 1 | u 0 = 1 )) ( P ( y 0 | x 0 = 0 ) , P ( y 0 | x 0 = 1 )) x 0 → y 0 u 0 u 0 � � ? x 0 � � � ( P ( y 0 y 1 � u 0 | u 1 = 0 ) , P ( y 0 y 1 � u 0 | u 1 = 1 )) ( P ( y 1 | x 1 = 0 ) , P ( y 1 | x 1 = 1 )) x 1 → y 1 u 1 � u 1 ? ? x 1 � probability pair variable boolean variable (bit)

  8. A closer look at successive cancellation � � ( P ( y 0 y 1 | u 0 = 0 ) , P ( y 0 y 1 | u 0 = 1 )) ( P ( y 0 | x 0 = 0 ) , P ( y 0 | x 0 = 1 )) x 0 → y 0 u 0 u 0 � � ? x 0 � � � ( P ( y 0 y 1 � u 0 | u 1 = 0 ) , P ( y 0 y 1 � u 0 | u 1 = 1 )) ( P ( y 1 | x 1 = 0 ) , P ( y 1 | x 1 = 1 )) x 1 → y 1 u 1 � u 1 ? ? x 1 � probability pair variable boolean variable (bit)

  9. A closer look at successive cancellation � � ( P ( y 0 y 1 | u 0 = 0 ) , P ( y 0 y 1 | u 0 = 1 )) ( P ( y 0 | x 0 = 0 ) , P ( y 0 | x 0 = 1 )) x 0 → y 0 u 0 u 0 � � ? x 0 � � � ( P ( y 0 y 1 � u 0 | u 1 = 0 ) , P ( y 0 y 1 � u 0 | u 1 = 1 )) ( P ( y 1 | x 1 = 0 ) , P ( y 1 | x 1 = 1 )) x 1 → y 1 u 1 � u 1 � ? x 1 � probability pair variable boolean variable (bit)

  10. A closer look at successive cancellation � � ( P ( y 0 y 1 | u 0 = 0 ) , P ( y 0 y 1 | u 0 = 1 )) ( P ( y 0 | x 0 = 0 ) , P ( y 0 | x 0 = 1 )) x 0 → y 0 u 0 u 0 � � ? x 0 � � � ( P ( y 0 y 1 � u 0 | u 1 = 0 ) , P ( y 0 y 1 � u 0 | u 1 = 1 )) ( P ( y 1 | x 1 = 0 ) , P ( y 1 | x 1 = 1 )) x 1 → y 1 u 1 � u 1 � ? x 1 � probability pair variable boolean variable (bit)

  11. A closer look at successive cancellation � � ( P ( y 0 y 1 | u 0 = 0 ) , P ( y 0 y 1 | u 0 = 1 )) ( P ( y 0 | x 0 = 0 ) , P ( y 0 | x 0 = 1 )) x 0 → y 0 u 0 u 0 � � � x 0 � � � ( P ( y 0 y 1 � u 0 | u 1 = 0 ) , P ( y 0 y 1 � u 0 | u 1 = 1 )) ( P ( y 1 | x 1 = 0 ) , P ( y 1 | x 1 = 1 )) x 1 → y 1 u 1 � u 1 � � x 1 � probability pair variable boolean variable (bit)

  12. A closer look at successive cancellation � � ( P ( y 0 y 1 | u 0 = 0 ) , P ( y 0 y 1 | u 0 = 1 )) ( P ( y 0 | x 0 = 0 ) , P ( y 0 | x 0 = 1 )) x 0 → y 0 u 0 u 0 � � � x 0 � � � ( P ( y 0 y 1 � u 0 | u 1 = 0 ) , P ( y 0 y 1 � u 0 | u 1 = 1 )) ( P ( y 1 | x 1 = 0 ) , P ( y 1 | x 1 = 1 )) x 1 → y 1 u 1 � u 1 � � x 1 � probability pair variable boolean variable (bit)

  13. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 Key point u 5 y 5 u 6 y 6 The memory u 7 y 7 needed to hold u 8 y 8 the variables at u 9 y 9 level t is O ( n /2 t ) . u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  14. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 Key point u 5 y 5 u 6 y 6 The memory u 7 y 7 needed to hold u 8 y 8 the variables at u 9 y 9 level t is O ( n /2 t ) . u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  15. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 Key point u 5 y 5 u 6 y 6 The memory u 7 y 7 needed to hold u 8 y 8 the variables at u 9 y 9 level t is O ( n /2 t ) . u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  16. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 Key point u 5 y 5 u 6 y 6 The memory u 7 y 7 needed to hold u 8 y 8 the variables at u 9 y 9 level t is O ( n /2 t ) . u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  17. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 Key point u 5 y 5 u 6 y 6 The memory u 7 y 7 needed to hold u 8 y 8 the variables at u 9 y 9 level t is O ( n /2 t ) . u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  18. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 Key point u 5 y 5 u 6 y 6 The memory u 7 y 7 needed to hold u 8 y 8 the variables at u 9 y 9 level t is O ( n /2 t ) . u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  19. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 Key point u 5 y 5 u 6 y 6 The memory u 7 y 7 needed to hold u 8 y 8 the variables at u 9 y 9 level t is O ( n /2 t ) . u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  20. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 Key point u 5 y 5 u 6 y 6 The memory u 7 y 7 needed to hold u 8 y 8 the variables at u 9 y 9 level t is O ( n /2 t ) . u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  21. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 Key point u 5 y 5 u 6 y 6 The memory u 7 y 7 needed to hold u 8 y 8 the variables at u 9 y 9 level t is O ( n /2 t ) . u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  22. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 Key point u 5 y 5 u 6 y 6 The memory u 7 y 7 needed to hold u 8 y 8 the variables at u 9 y 9 level t is O ( n /2 t ) . u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  23. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 Key point u 5 y 5 u 6 y 6 The memory u 7 y 7 needed to hold u 8 y 8 the variables at u 9 y 9 level t is O ( n /2 t ) . u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  24. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 Key point u 5 y 5 u 6 y 6 The memory u 7 y 7 needed to hold u 8 y 8 the variables at u 9 y 9 level t is O ( n /2 t ) . u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  25. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 Key point u 5 y 5 u 6 y 6 The memory u 7 y 7 needed to hold u 8 y 8 the variables at u 9 y 9 level t is O ( n /2 t ) . u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  26. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 Key point u 5 y 5 u 6 y 6 The memory u 7 y 7 needed to hold u 8 y 8 the variables at u 9 y 9 level t is O ( n /2 t ) . u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  27. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 Key point u 5 y 5 u 6 y 6 The memory u 7 y 7 needed to hold u 8 y 8 the variables at u 9 y 9 level t is O ( n /2 t ) . u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  28. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 Key point u 5 y 5 u 6 y 6 The memory u 7 y 7 needed to hold u 8 y 8 the variables at u 9 y 9 level t is O ( n /2 t ) . u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  29. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 Key point u 5 y 5 u 6 y 6 The memory u 7 y 7 needed to hold u 8 y 8 the variables at u 9 y 9 level t is O ( n /2 t ) . u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  30. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 Key point u 5 y 5 u 6 y 6 The memory u 7 y 7 needed to hold u 8 y 8 the variables at u 9 y 9 level t is O ( n /2 t ) . u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  31. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 Key point u 5 y 5 u 6 y 6 The memory u 7 y 7 needed to hold u 8 y 8 the variables at u 9 y 9 level t is O ( n /2 t ) . u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  32. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 Key point u 5 y 5 u 6 y 6 The memory u 7 y 7 needed to hold u 8 y 8 the variables at u 9 y 9 level t is O ( n /2 t ) . u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  33. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 Key point u 5 y 5 u 6 y 6 The memory u 7 y 7 needed to hold u 8 y 8 the variables at u 9 y 9 level t is O ( n /2 t ) . u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  34. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 Key point u 5 y 5 u 6 y 6 The memory u 7 y 7 needed to hold u 8 y 8 the variables at u 9 y 9 level t is O ( n /2 t ) . u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  35. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 Key point u 5 y 5 u 6 y 6 The memory u 7 y 7 needed to hold u 8 y 8 the variables at u 9 y 9 level t is O ( n /2 t ) . u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  36. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 Key point u 5 y 5 u 6 y 6 The memory u 7 y 7 needed to hold u 8 y 8 the variables at u 9 y 9 level t is O ( n /2 t ) . u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  37. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 Key point u 5 y 5 u 6 y 6 The memory u 7 y 7 needed to hold u 8 y 8 the variables at u 9 y 9 level t is O ( n /2 t ) . u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  38. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 Key point u 5 y 5 u 6 y 6 The memory u 7 y 7 needed to hold u 8 y 8 the variables at u 9 y 9 level t is O ( n /2 t ) . u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  39. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 Key point u 5 y 5 u 6 y 6 The memory u 7 y 7 needed to hold u 8 y 8 the variables at u 9 y 9 level t is O ( n /2 t ) . u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  40. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 Key point u 5 y 5 u 6 y 6 The memory u 7 y 7 needed to hold u 8 y 8 the variables at u 9 y 9 level t is O ( n /2 t ) . u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  41. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 Key point u 5 y 5 u 6 y 6 The memory u 7 y 7 needed to hold u 8 y 8 the variables at u 9 y 9 level t is O ( n /2 t ) . u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  42. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 Key point u 5 y 5 u 6 y 6 The memory u 7 y 7 needed to hold u 8 y 8 the variables at u 9 y 9 level t is O ( n /2 t ) . u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  43. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 Key point u 5 y 5 u 6 y 6 The memory u 7 y 7 needed to hold u 8 y 8 the variables at u 9 y 9 level t is O ( n /2 t ) . u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  44. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 Key point u 5 y 5 u 6 y 6 The memory u 7 y 7 needed to hold u 8 y 8 the variables at u 9 y 9 level t is O ( n /2 t ) . u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  45. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 u 5 y 5 Key point u 6 y 6 Level t is written u 7 y 7 to once every u 8 y 8 O ( 2 m − t ) stages. u 9 y 9 u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  46. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 u 5 y 5 Key point u 6 y 6 Level t is written u 7 y 7 to once every u 8 y 8 O ( 2 m − t ) stages. u 9 y 9 u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  47. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 u 5 y 5 Key point u 6 y 6 Level t is written u 7 y 7 to once every u 8 y 8 O ( 2 m − t ) stages. u 9 y 9 u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  48. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 u 5 y 5 Key point u 6 y 6 Level t is written u 7 y 7 to once every u 8 y 8 O ( 2 m − t ) stages. u 9 y 9 u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  49. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 u 5 y 5 Key point u 6 y 6 Level t is written u 7 y 7 to once every u 8 y 8 O ( 2 m − t ) stages. u 9 y 9 u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  50. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 u 5 y 5 Key point u 6 y 6 Level t is written u 7 y 7 to once every u 8 y 8 O ( 2 m − t ) stages. u 9 y 9 u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  51. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 u 5 y 5 Key point u 6 y 6 Level t is written u 7 y 7 to once every u 8 y 8 O ( 2 m − t ) stages. u 9 y 9 u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  52. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 u 5 y 5 Key point u 6 y 6 Level t is written u 7 y 7 to once every u 8 y 8 O ( 2 m − t ) stages. u 9 y 9 u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  53. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 u 5 y 5 Key point u 6 y 6 Level t is written u 7 y 7 to once every u 8 y 8 O ( 2 m − t ) stages. u 9 y 9 u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  54. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 u 5 y 5 Key point u 6 y 6 Level t is written u 7 y 7 to once every u 8 y 8 O ( 2 m − t ) stages. u 9 y 9 u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  55. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 u 5 y 5 Key point u 6 y 6 Level t is written u 7 y 7 to once every u 8 y 8 O ( 2 m − t ) stages. u 9 y 9 u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  56. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 u 5 y 5 Key point u 6 y 6 Level t is written u 7 y 7 to once every u 8 y 8 O ( 2 m − t ) stages. u 9 y 9 u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  57. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 u 5 y 5 Key point u 6 y 6 Level t is written u 7 y 7 to once every u 8 y 8 O ( 2 m − t ) stages. u 9 y 9 u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  58. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 u 5 y 5 Key point u 6 y 6 Level t is written u 7 y 7 to once every u 8 y 8 O ( 2 m − t ) stages. u 9 y 9 u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  59. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 u 5 y 5 Key point u 6 y 6 Level t is written u 7 y 7 to once every u 8 y 8 O ( 2 m − t ) stages. u 9 y 9 u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  60. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 u 5 y 5 Key point u 6 y 6 Level t is written u 7 y 7 to once every u 8 y 8 O ( 2 m − t ) stages. u 9 y 9 u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  61. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 u 5 y 5 Key point u 6 y 6 Level t is written u 7 y 7 to once every u 8 y 8 O ( 2 m − t ) stages. u 9 y 9 u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  62. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 u 5 y 5 Key point u 6 y 6 Level t is written u 7 y 7 to once every u 8 y 8 O ( 2 m − t ) stages. u 9 y 9 u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  63. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 u 5 y 5 Key point u 6 y 6 Level t is written u 7 y 7 to once every u 8 y 8 O ( 2 m − t ) stages. u 9 y 9 u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  64. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 u 5 y 5 Key point u 6 y 6 Level t is written u 7 y 7 to once every u 8 y 8 O ( 2 m − t ) stages. u 9 y 9 u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  65. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 u 5 y 5 Key point u 6 y 6 Level t is written u 7 y 7 to once every u 8 y 8 O ( 2 m − t ) stages. u 9 y 9 u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  66. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 u 5 y 5 Key point u 6 y 6 Level t is written u 7 y 7 to once every u 8 y 8 O ( 2 m − t ) stages. u 9 y 9 u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  67. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 u 5 y 5 Key point u 6 y 6 Level t is written u 7 y 7 to once every u 8 y 8 O ( 2 m − t ) stages. u 9 y 9 u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  68. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 u 5 y 5 Key point u 6 y 6 Level t is written u 7 y 7 to once every u 8 y 8 O ( 2 m − t ) stages. u 9 y 9 u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  69. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 u 5 y 5 Key point u 6 y 6 Level t is written u 7 y 7 to once every u 8 y 8 O ( 2 m − t ) stages. u 9 y 9 u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  70. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 u 5 y 5 Key point u 6 y 6 Level t is written u 7 y 7 to once every u 8 y 8 O ( 2 m − t ) stages. u 9 y 9 u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  71. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 u 5 y 5 Key point u 6 y 6 Level t is written u 7 y 7 to once every u 8 y 8 O ( 2 m − t ) stages. u 9 y 9 u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  72. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 u 5 y 5 Key point u 6 y 6 Level t is written u 7 y 7 to once every u 8 y 8 O ( 2 m − t ) stages. u 9 y 9 u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

  73. A larger example 4 3 2 1 0 u 0 y 0 u 1 y 1 u 2 y 2 u 3 y 3 u 4 y 4 u 5 y 5 Key point u 6 y 6 Level t is written u 7 y 7 to once every u 8 y 8 O ( 2 m − t ) stages. u 9 y 9 u 10 y 10 y 11 u 11 u 12 y 12 u 13 y 13 u 14 y 14 u 15 y 15

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