MinT - Best Known (21−8, 21, s)-Nets in Base 64

Best Known (21−8, 21, s)-Nets in Base 64

(21−8, 21, 4175)-Net over F₆₄ — Constructive and digital

Digital (13, 21, 4175)-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, 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]
58. digital (0, 1, 65)-net over F₆₄ (see above)
59. digital (0, 1, 65)-net over F₆₄ (see above)
60. digital (0, 1, 65)-net over F₆₄ (see above)
61. digital (0, 2, 65)-net over F₆₄, using
  - kÂ =Â 2 construction [i]
62. digital (0, 2, 65)-net over F₆₄ (see above)
63. digital (0, 4, 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]
64. digital (1, 9, 80)-net over F₆₄, using
  - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
      - a function field by SÃ©mirat [i]

(21−8, 21, 14068)-Net over F₆₄ — Digital

Digital (13, 21, 14068)-net over F₆₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(21−8, 21, 16384)-Net in Base 64 — Constructive

(13, 21, 16384)-net in base 64, using

64¹ times duplication [i] based on (12, 20, 16384)-net in base 64, using
- base change [i] based on digital (7, 15, 16384)-net over F₂₅₆, using
  - net defined by OOA [i] based on linear OOA(256¹⁵, 16384, F₂₅₆, 8, 8) (dual of [(16384, 8), 131057, 9]-NRT-code), using
    - OA 4-folding and stacking [i] based on linear OA(256¹⁵, 65536, F₂₅₆, 8) (dual of [65536, 65521, 9]-code), using
      - an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]

(21−8, 21, 21846)-Net in Base 64

(13, 21, 21846)-net in base 64, using

64¹ times duplication [i] based on (12, 20, 21846)-net in base 64, using
- base change [i] based on digital (7, 15, 21846)-net over F₂₅₆, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(256¹⁵, 21846, F₂₅₆, 3, 8) (dual of [(21846, 3), 65523, 9]-NRT-code), using
    - OOA 3-folding [i] based on linear OA(256¹⁵, 65538, F₂₅₆, 8) (dual of [65538, 65523, 9]-code), using
      - constructionÂ X applied to C_e(7) âŠ‚ C_e(6) [i] based on
        linear OA(256¹⁵, 65536, F₂₅₆, 8) (dual of [65536, 65521, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
        linear OA(256¹³, 65536, F₂₅₆, 7) (dual of [65536, 65523, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(256⁰, 2, F₂₅₆, 0) (dual of [2, 2, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁰, s, F₂₅₆, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(21−8, 21, large)-Net in Base 64 — Upper bound on s

There is no (13, 21, large)-net in base 64, because

6 times m-reduction [i] would yield (13, 15, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]