MinT - Best Known (5, 10, s)-Nets in Base 64

Best Known (5, 10, s)-Nets in Base 64

(5, 10, 4160)-Net over F₆₄ — Constructive and digital

Digital (5, 10, 4160)-net over F₆₄, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 0, 65)-net over F₆₄, using
  - s-reduction based on digital (0, 0, s)-net over F₆₄ with arbitrarily large s, using
    - arbitrary set of 64⁰ points [i]
2. digital (0, 0, 65)-net over F₆₄ (see above)
3. digital (0, 0, 65)-net over F₆₄ (see above)
4. digital (0, 0, 65)-net over F₆₄ (see above)
5. digital (0, 0, 65)-net over F₆₄ (see above)
6. digital (0, 0, 65)-net over F₆₄ (see above)
7. digital (0, 0, 65)-net over F₆₄ (see above)
8. digital (0, 0, 65)-net over F₆₄ (see above)
9. digital (0, 0, 65)-net over F₆₄ (see above)
10. digital (0, 0, 65)-net over F₆₄ (see above)
11. digital (0, 0, 65)-net over F₆₄ (see above)
12. digital (0, 0, 65)-net over F₆₄ (see above)
13. digital (0, 0, 65)-net over F₆₄ (see above)
14. digital (0, 0, 65)-net over F₆₄ (see above)
15. digital (0, 0, 65)-net over F₆₄ (see above)
16. digital (0, 0, 65)-net over F₆₄ (see above)
17. digital (0, 0, 65)-net over F₆₄ (see above)
18. digital (0, 0, 65)-net over F₆₄ (see above)
19. digital (0, 0, 65)-net over F₆₄ (see above)
20. digital (0, 0, 65)-net over F₆₄ (see above)
21. digital (0, 0, 65)-net over F₆₄ (see above)
22. digital (0, 0, 65)-net over F₆₄ (see above)
23. digital (0, 0, 65)-net over F₆₄ (see above)
24. digital (0, 0, 65)-net over F₆₄ (see above)
25. digital (0, 0, 65)-net over F₆₄ (see above)
26. digital (0, 0, 65)-net over F₆₄ (see above)
27. digital (0, 0, 65)-net over F₆₄ (see above)
28. digital (0, 0, 65)-net over F₆₄ (see above)
29. digital (0, 0, 65)-net over F₆₄ (see above)
30. digital (0, 0, 65)-net over F₆₄ (see above)
31. digital (0, 0, 65)-net over F₆₄ (see above)
32. digital (0, 0, 65)-net over F₆₄ (see above)
33. digital (0, 0, 65)-net over F₆₄ (see above)
34. digital (0, 0, 65)-net over F₆₄ (see above)
35. digital (0, 0, 65)-net over F₆₄ (see above)
36. digital (0, 0, 65)-net over F₆₄ (see above)
37. digital (0, 0, 65)-net over F₆₄ (see above)
38. digital (0, 0, 65)-net over F₆₄ (see above)
39. digital (0, 0, 65)-net over F₆₄ (see above)
40. digital (0, 0, 65)-net over F₆₄ (see above)
41. digital (0, 0, 65)-net over F₆₄ (see above)
42. digital (0, 0, 65)-net over F₆₄ (see above)
43. digital (0, 0, 65)-net over F₆₄ (see above)
44. digital (0, 0, 65)-net over F₆₄ (see above)
45. digital (0, 0, 65)-net over F₆₄ (see above)
46. digital (0, 0, 65)-net over F₆₄ (see above)
47. digital (0, 0, 65)-net over F₆₄ (see above)
48. digital (0, 0, 65)-net over F₆₄ (see above)
49. digital (0, 0, 65)-net over F₆₄ (see above)
50. digital (0, 0, 65)-net over F₆₄ (see above)
51. digital (0, 0, 65)-net over F₆₄ (see above)
52. digital (0, 0, 65)-net over F₆₄ (see above)
53. digital (0, 0, 65)-net over F₆₄ (see above)
54. digital (0, 0, 65)-net over F₆₄ (see above)
55. digital (0, 0, 65)-net over F₆₄ (see above)
56. digital (0, 0, 65)-net over F₆₄ (see above)
57. digital (0, 0, 65)-net over F₆₄ (see above)
58. digital (0, 0, 65)-net over F₆₄ (see above)
59. digital (0, 0, 65)-net over F₆₄ (see above)
60. digital (0, 1, 65)-net over F₆₄, using
  - s-reduction based on digital (0, 1, s)-net over F₆₄ with arbitrarily large s, using
    - construction for strength kÂ =Â 1 [i]
61. digital (0, 1, 65)-net over F₆₄ (see above)
62. digital (0, 1, 65)-net over F₆₄ (see above)
63. digital (0, 2, 65)-net over F₆₄, using
  - kÂ =Â 2 construction [i]
64. digital (0, 5, 65)-net over F₆₄, using
  - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
      - the rational function field F₆₄(x) [i]
    - Niederreiter sequence [i]

(5, 10, 7561)-Net over F₆₄ — Digital

Digital (5, 10, 7561)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64¹⁰, 7561, F₆₄, 5) (dual of [7561, 7551, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(64¹⁰, 8066, F₆₄, 5) (dual of [8066, 8056, 6]-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OA(64³, 4033, F₆₄, 2) (dual of [4033, 4030, 3]-code), using
      - discarding factors / shortening the dual code based on linear OA(64³, 4161, F₆₄, 2) (dual of [4161, 4158, 3]-code), using
        Hamming code H(3,64) [i]
    2. linear OA(64⁷, 4033, F₆₄, 5) (dual of [4033, 4026, 6]-code), using
      - a code by Danev and Olsson [i]

(5, 10, 32640)-Net in Base 64 — Constructive

(5, 10, 32640)-net in base 64, using

net defined by OOA [i] based on OOA(64¹⁰, 32640, S₆₄, 5, 5), using
- OOA 2-folding and stacking with additional row [i] based on OA(64¹⁰, 65281, S₆₄, 5), using
  - discarding parts of the base [i] based on linear OA(256⁷, 65281, F₂₅₆, 5) (dual of [65281, 65274, 6]-code), using
    - a code by Danev and Olsson [i]

(5, 10, 3012896)-Net in Base 64 — Upper bound on s

There is no (5, 10, 3012897)-net in base 64, because

1 times m-reduction [i] would yield (5, 9, 3012897)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 18014 401024 781680 > 64⁹ [i]

Best Known (5, 10, s)-Nets in Base 64

(5, 10, 4160)-Net over F64 — Constructive and digital

(5, 10, 7561)-Net over F64 — Digital

(5, 10, 32640)-Net in Base 64 — Constructive

(5, 10, 3012896)-Net in Base 64 — Upper bound on s

(5, 10, 4160)-Net over F₆₄ — Constructive and digital

(5, 10, 7561)-Net over F₆₄ — Digital