MinT - Best Known (41, 41+9, s)-Nets in Base 81

Best Known (41, 41+9, s)-Nets in Base 81

(41, 41+9, large)-Net over F₈₁ — Constructive and digital

Digital (41, 50, large)-net over F₈₁, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 0, 132860)-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, 132860)-net over F₈₁ (see above)
3. digital (0, 0, 132860)-net over F₈₁ (see above)
4. digital (0, 0, 132860)-net over F₈₁ (see above)
5. digital (0, 0, 132860)-net over F₈₁ (see above)
6. digital (0, 0, 132860)-net over F₈₁ (see above)
7. digital (0, 0, 132860)-net over F₈₁ (see above)
8. digital (0, 0, 132860)-net over F₈₁ (see above)
9. digital (0, 0, 132860)-net over F₈₁ (see above)
10. digital (0, 0, 132860)-net over F₈₁ (see above)
11. digital (0, 0, 132860)-net over F₈₁ (see above)
12. digital (0, 0, 132860)-net over F₈₁ (see above)
13. digital (0, 0, 132860)-net over F₈₁ (see above)
14. digital (0, 0, 132860)-net over F₈₁ (see above)
15. digital (0, 0, 132860)-net over F₈₁ (see above)
16. digital (0, 0, 132860)-net over F₈₁ (see above)
17. digital (0, 0, 132860)-net over F₈₁ (see above)
18. digital (0, 0, 132860)-net over F₈₁ (see above)
19. digital (0, 0, 132860)-net over F₈₁ (see above)
20. digital (0, 0, 132860)-net over F₈₁ (see above)
21. digital (0, 0, 132860)-net over F₈₁ (see above)
22. digital (0, 0, 132860)-net over F₈₁ (see above)
23. digital (0, 0, 132860)-net over F₈₁ (see above)
24. digital (0, 0, 132860)-net over F₈₁ (see above)
25. digital (0, 0, 132860)-net over F₈₁ (see above)
26. digital (0, 0, 132860)-net over F₈₁ (see above)
27. digital (0, 0, 132860)-net over F₈₁ (see above)
28. digital (0, 0, 132860)-net over F₈₁ (see above)
29. digital (0, 0, 132860)-net over F₈₁ (see above)
30. digital (0, 0, 132860)-net over F₈₁ (see above)
31. digital (0, 0, 132860)-net over F₈₁ (see above)
32. digital (0, 0, 132860)-net over F₈₁ (see above)
33. digital (0, 0, 132860)-net over F₈₁ (see above)
34. digital (0, 0, 132860)-net over F₈₁ (see above)
35. digital (0, 0, 132860)-net over F₈₁ (see above)
36. digital (0, 0, 132860)-net over F₈₁ (see above)
37. digital (0, 0, 132860)-net over F₈₁ (see above)
38. digital (0, 0, 132860)-net over F₈₁ (see above)
39. digital (0, 0, 132860)-net over F₈₁ (see above)
40. digital (0, 0, 132860)-net over F₈₁ (see above)
41. digital (0, 0, 132860)-net over F₈₁ (see above)
42. digital (0, 0, 132860)-net over F₈₁ (see above)
43. digital (0, 0, 132860)-net over F₈₁ (see above)
44. digital (0, 0, 132860)-net over F₈₁ (see above)
45. digital (0, 0, 132860)-net over F₈₁ (see above)
46. digital (0, 0, 132860)-net over F₈₁ (see above)
47. digital (0, 0, 132860)-net over F₈₁ (see above)
48. digital (0, 0, 132860)-net over F₈₁ (see above)
49. digital (0, 0, 132860)-net over F₈₁ (see above)
50. digital (0, 0, 132860)-net over F₈₁ (see above)
51. digital (0, 0, 132860)-net over F₈₁ (see above)
52. digital (0, 0, 132860)-net over F₈₁ (see above)
53. digital (0, 0, 132860)-net over F₈₁ (see above)
54. digital (0, 0, 132860)-net over F₈₁ (see above)
55. digital (0, 0, 132860)-net over F₈₁ (see above)
56. digital (0, 0, 132860)-net over F₈₁ (see above)
57. digital (0, 0, 132860)-net over F₈₁ (see above)
58. digital (0, 0, 132860)-net over F₈₁ (see above)
59. digital (0, 0, 132860)-net over F₈₁ (see above)
60. digital (0, 0, 132860)-net over F₈₁ (see above)
61. digital (0, 0, 132860)-net over F₈₁ (see above)
62. digital (0, 0, 132860)-net over F₈₁ (see above)
63. digital (0, 0, 132860)-net over F₈₁ (see above)
64. digital (0, 0, 132860)-net over F₈₁ (see above)
65. digital (0, 0, 132860)-net over F₈₁ (see above)
66. digital (0, 0, 132860)-net over F₈₁ (see above)
67. digital (0, 0, 132860)-net over F₈₁ (see above)
68. digital (0, 0, 132860)-net over F₈₁ (see above)
69. digital (0, 0, 132860)-net over F₈₁ (see above)
70. digital (0, 0, 132860)-net over F₈₁ (see above)
71. digital (0, 0, 132860)-net over F₈₁ (see above)
72. digital (0, 0, 132860)-net over F₈₁ (see above)
73. digital (0, 1, 132860)-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]
74. digital (0, 1, 132860)-net over F₈₁ (see above)
75. digital (0, 1, 132860)-net over F₈₁ (see above)
76. digital (0, 1, 132860)-net over F₈₁ (see above)
77. digital (0, 1, 132860)-net over F₈₁ (see above)
78. digital (2, 4, 132860)-net over F₈₁, using
  - s-reduction based on digital (2, 4, 538084)-net over F₈₁, using
    - kÂ =Â 2 construction [i]
79. digital (3, 6, 132860)-net over F₈₁, using
  - s-reduction based on digital (3, 6, 538248)-net over F₈₁, using
    - net defined by OOA [i] based on linear OOA(81⁶, 538248, F₈₁, 3, 3) (dual of [(538248, 3), 1614738, 4]-NRT-code), using
      - appending kth column [i] based on linear OOA(81⁶, 538248, F₈₁, 2, 3) (dual of [(538248, 2), 1076490, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(81⁶, 538248, F₈₁, 3) (dual of [538248, 538242, 4]-code or 538248-cap in PG(5,81)), using
        large caps in PG(5,Â b) [i]
80. digital (6, 10, 132860)-net over F₈₁, using
  - s-reduction based on digital (6, 10, 265722)-net over F₈₁, using
    - net defined by OOA [i] based on linear OOA(81¹⁰, 265722, F₈₁, 4, 4) (dual of [(265722, 4), 1062878, 5]-NRT-code), using
      - OA 2-folding and stacking [i] based on linear OA(81¹⁰, 531444, F₈₁, 4) (dual of [531444, 531434, 5]-code), using
        constructionÂ X applied to C_e(3) âŠ‚ C_e(2) [i] based on
        linear OA(81¹⁰, 531441, F₈₁, 4) (dual of [531441, 531431, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(81⁷, 531441, F₈₁, 3) (dual of [531441, 531434, 4]-code or 531441-cap in PG(6,81)), using an extension C_e(2) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
        linear OA(81⁰, 3, F₈₁, 0) (dual of [3, 3, 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]
81. digital (16, 25, 132860)-net over F₈₁, using
  - net defined by OOA [i] based on linear OOA(81²⁵, 132860, F₈₁, 9, 9) (dual of [(132860, 9), 1195715, 10]-NRT-code), using
    - OOA 4-folding and stacking with additional row [i] based on linear OA(81²⁵, 531441, F₈₁, 9) (dual of [531441, 531416, 10]-code), using
      - an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]

(41, 41+9, large)-Net in Base 81 — Upper bound on s

There is no (41, 50, large)-net in base 81, because

7 times m-reduction [i] would yield (41, 43, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]

Best Known (41, 41+9, s)-Nets in Base 81

(41, 41+9, large)-Net over F81 — Constructive and digital

(41, 41+9, large)-Net in Base 81 — Upper bound on s

(41, 41+9, large)-Net over F₈₁ — Constructive and digital