MinT - Best Known (12, 12+8, s)-Nets in Base 81

Best Known (12, 12+8, s)-Nets in Base 81

(12, 12+8, 6642)-Net over F₈₁ — Constructive and digital

Digital (12, 20, 6642)-net over F₈₁, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 0, 82)-net over F₈₁, using
  - s-reduction based on digital (0, 0, s)-net over F₈₁ with arbitrarily large s, using
    - arbitrary set of 81⁰ points [i]
2. digital (0, 0, 82)-net over F₈₁ (see above)
3. digital (0, 0, 82)-net over F₈₁ (see above)
4. digital (0, 0, 82)-net over F₈₁ (see above)
5. digital (0, 0, 82)-net over F₈₁ (see above)
6. digital (0, 0, 82)-net over F₈₁ (see above)
7. digital (0, 0, 82)-net over F₈₁ (see above)
8. digital (0, 0, 82)-net over F₈₁ (see above)
9. digital (0, 0, 82)-net over F₈₁ (see above)
10. digital (0, 0, 82)-net over F₈₁ (see above)
11. digital (0, 0, 82)-net over F₈₁ (see above)
12. digital (0, 0, 82)-net over F₈₁ (see above)
13. digital (0, 0, 82)-net over F₈₁ (see above)
14. digital (0, 0, 82)-net over F₈₁ (see above)
15. digital (0, 0, 82)-net over F₈₁ (see above)
16. digital (0, 0, 82)-net over F₈₁ (see above)
17. digital (0, 0, 82)-net over F₈₁ (see above)
18. digital (0, 0, 82)-net over F₈₁ (see above)
19. digital (0, 0, 82)-net over F₈₁ (see above)
20. digital (0, 0, 82)-net over F₈₁ (see above)
21. digital (0, 0, 82)-net over F₈₁ (see above)
22. digital (0, 0, 82)-net over F₈₁ (see above)
23. digital (0, 0, 82)-net over F₈₁ (see above)
24. digital (0, 0, 82)-net over F₈₁ (see above)
25. digital (0, 0, 82)-net over F₈₁ (see above)
26. digital (0, 0, 82)-net over F₈₁ (see above)
27. digital (0, 0, 82)-net over F₈₁ (see above)
28. digital (0, 0, 82)-net over F₈₁ (see above)
29. digital (0, 0, 82)-net over F₈₁ (see above)
30. digital (0, 0, 82)-net over F₈₁ (see above)
31. digital (0, 0, 82)-net over F₈₁ (see above)
32. digital (0, 0, 82)-net over F₈₁ (see above)
33. digital (0, 0, 82)-net over F₈₁ (see above)
34. digital (0, 0, 82)-net over F₈₁ (see above)
35. digital (0, 0, 82)-net over F₈₁ (see above)
36. digital (0, 0, 82)-net over F₈₁ (see above)
37. digital (0, 0, 82)-net over F₈₁ (see above)
38. digital (0, 0, 82)-net over F₈₁ (see above)
39. digital (0, 0, 82)-net over F₈₁ (see above)
40. digital (0, 0, 82)-net over F₈₁ (see above)
41. digital (0, 0, 82)-net over F₈₁ (see above)
42. digital (0, 0, 82)-net over F₈₁ (see above)
43. digital (0, 0, 82)-net over F₈₁ (see above)
44. digital (0, 0, 82)-net over F₈₁ (see above)
45. digital (0, 0, 82)-net over F₈₁ (see above)
46. digital (0, 0, 82)-net over F₈₁ (see above)
47. digital (0, 0, 82)-net over F₈₁ (see above)
48. digital (0, 0, 82)-net over F₈₁ (see above)
49. digital (0, 0, 82)-net over F₈₁ (see above)
50. digital (0, 0, 82)-net over F₈₁ (see above)
51. digital (0, 0, 82)-net over F₈₁ (see above)
52. digital (0, 0, 82)-net over F₈₁ (see above)
53. digital (0, 0, 82)-net over F₈₁ (see above)
54. digital (0, 0, 82)-net over F₈₁ (see above)
55. digital (0, 0, 82)-net over F₈₁ (see above)
56. digital (0, 0, 82)-net over F₈₁ (see above)
57. digital (0, 0, 82)-net over F₈₁ (see above)
58. digital (0, 0, 82)-net over F₈₁ (see above)
59. digital (0, 0, 82)-net over F₈₁ (see above)
60. digital (0, 0, 82)-net over F₈₁ (see above)
61. digital (0, 0, 82)-net over F₈₁ (see above)
62. digital (0, 0, 82)-net over F₈₁ (see above)
63. digital (0, 0, 82)-net over F₈₁ (see above)
64. digital (0, 0, 82)-net over F₈₁ (see above)
65. digital (0, 0, 82)-net over F₈₁ (see above)
66. digital (0, 0, 82)-net over F₈₁ (see above)
67. digital (0, 0, 82)-net over F₈₁ (see above)
68. digital (0, 0, 82)-net over F₈₁ (see above)
69. digital (0, 0, 82)-net over F₈₁ (see above)
70. digital (0, 0, 82)-net over F₈₁ (see above)
71. digital (0, 0, 82)-net over F₈₁ (see above)
72. digital (0, 0, 82)-net over F₈₁ (see above)
73. digital (0, 0, 82)-net over F₈₁ (see above)
74. digital (0, 1, 82)-net over F₈₁, using
  - s-reduction based on digital (0, 1, s)-net over F₈₁ with arbitrarily large s, using
    - construction for strength kÂ =Â 1 [i]
75. digital (0, 1, 82)-net over F₈₁ (see above)
76. digital (0, 1, 82)-net over F₈₁ (see above)
77. digital (0, 1, 82)-net over F₈₁ (see above)
78. digital (0, 2, 82)-net over F₈₁, using
  - kÂ =Â 2 construction [i]
79. digital (0, 2, 82)-net over F₈₁ (see above)
80. digital (0, 4, 82)-net over F₈₁, using
  - net from sequence [i] based on digital (0, 81)-sequence over F₈₁, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 0 and N(F) ≥ 82, using
      - the rational function field F₈₁(x) [i]
    - Niederreiter sequence [i]
81. digital (0, 8, 82)-net over F₈₁, using
  - net from sequence [i] based on digital (0, 81)-sequence over F₈₁ (see above)

(12, 12+8, 11989)-Net over F₈₁ — Digital

Digital (12, 20, 11989)-net over F₈₁, using

Gilbertâ€“VarÅ¡amov bound [i]

(12, 12+8, large)-Net in Base 81 — Upper bound on s

There is no (12, 20, large)-net in base 81, because

6 times m-reduction [i] would yield (12, 14, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]

Best Known (12, 12+8, s)-Nets in Base 81

(12, 12+8, 6642)-Net over F81 — Constructive and digital

(12, 12+8, 11989)-Net over F81 — Digital

(12, 12+8, large)-Net in Base 81 — Upper bound on s

(12, 12+8, 6642)-Net over F₈₁ — Constructive and digital

(12, 12+8, 11989)-Net over F₈₁ — Digital