MinT - Best Known (9, 9+7, s)-Nets in Base 81

Best Known (9, 9+7, s)-Nets in Base 81

(9, 9+7, 6642)-Net over F₈₁ — Constructive and digital

Digital (9, 16, 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, 0, 82)-net over F₈₁ (see above)
75. 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]
76. digital (0, 1, 82)-net over F₈₁ (see above)
77. digital (0, 1, 82)-net over F₈₁ (see above)
78. digital (0, 1, 82)-net over F₈₁ (see above)
79. digital (0, 2, 82)-net over F₈₁, using
  - kÂ =Â 2 construction [i]
80. digital (0, 3, 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, 7, 82)-net over F₈₁, using
  - net from sequence [i] based on digital (0, 81)-sequence over F₈₁ (see above)

(9, 9+7, 6645)-Net over F₈₁ — Digital

Digital (9, 16, 6645)-net over F₈₁, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(81¹⁶, 6645, F₈₁, 7) (dual of [6645, 6629, 8]-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OA(81³, 82, F₈₁, 3) (dual of [82, 79, 4]-code or 82-arc in PG(2,81) or 82-cap in PG(2,81)), using
    - extended Reedâ€“Solomon code RS_e(79,81) [i]
    - oval in PG(2,Â 81) [i]
  2. linear OA(81¹³, 6563, F₈₁, 7) (dual of [6563, 6550, 8]-code), using
    - constructionÂ X applied to C_e(6) âŠ‚ C_e(5) [i] based on
      1. linear OA(81¹³, 6561, F₈₁, 7) (dual of [6561, 6548, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
      2. linear OA(81¹¹, 6561, F₈₁, 6) (dual of [6561, 6550, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
      3. linear OA(81⁰, 2, F₈₁, 0) (dual of [2, 2, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(81⁰, s, F₈₁, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(9, 9+7, large)-Net in Base 81 — Upper bound on s

There is no (9, 16, large)-net in base 81, because

5 times m-reduction [i] would yield (9, 11, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]

Best Known (9, 9+7, s)-Nets in Base 81

(9, 9+7, 6642)-Net over F81 — Constructive and digital

(9, 9+7, 6645)-Net over F81 — Digital

(9, 9+7, large)-Net in Base 81 — Upper bound on s

(9, 9+7, 6642)-Net over F₈₁ — Constructive and digital

(9, 9+7, 6645)-Net over F₈₁ — Digital