MinT - Best Known (37, 47, s)-Nets in Base 64

Best Known (37, 47, s)-Nets in Base 64

(37, 47, 1681880)-Net over F₆₄ — Constructive and digital

Digital (37, 47, 1681880)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. 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]
2. digital (27, 37, 1677720)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64³⁷, 1677720, F₆₄, 10, 10) (dual of [(1677720, 10), 16777163, 11]-NRT-code), using
    - OA 5-folding and stacking [i] based on linear OA(64³⁷, 8388600, F₆₄, 10) (dual of [8388600, 8388563, 11]-code), using
      - discarding factors / shortening the dual code based on linear OA(64³⁷, large, F₆₄, 10) (dual of [large, large−37, 11]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(37, 47, 1710360)-Net in Base 64 — Constructive

(37, 47, 1710360)-net in base 64, using

(u,Â u+v)-construction [i] based on
1. (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]
2. digital (27, 37, 1677720)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64³⁷, 1677720, F₆₄, 10, 10) (dual of [(1677720, 10), 16777163, 11]-NRT-code), using
    - OA 5-folding and stacking [i] based on linear OA(64³⁷, 8388600, F₆₄, 10) (dual of [8388600, 8388563, 11]-code), using
      - discarding factors / shortening the dual code based on linear OA(64³⁷, large, F₆₄, 10) (dual of [large, large−37, 11]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(37, 47, large)-Net over F₆₄ — Digital

Digital (37, 47, large)-net over F₆₄, using

3 times m-reduction [i] based on digital (37, 50, large)-net over F₆₄, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64⁵⁰, large, F₆₄, 13) (dual of [large, large−50, 14]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(64⁴⁹, large, F₆₄, 13) (dual of [large, large−49, 14]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(37, 47, large)-Net in Base 64 — Upper bound on s

There is no (37, 47, large)-net in base 64, because

8 times m-reduction [i] would yield (37, 39, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]

Best Known (37, 47, s)-Nets in Base 64

(37, 47, 1681880)-Net over F64 — Constructive and digital

(37, 47, 1710360)-Net in Base 64 — Constructive

(37, 47, large)-Net over F64 — Digital

(37, 47, large)-Net in Base 64 — Upper bound on s

(37, 47, 1681880)-Net over F₆₄ — Constructive and digital

(37, 47, large)-Net over F₆₄ — Digital