MinT - Best Known (57, 71, s)-Nets in Base 81

Best Known (57, 71, s)-Nets in Base 81

(57, 71, 1205049)-Net over F₈₁ — Constructive and digital

Digital (57, 71, 1205049)-net over F₈₁, using

(u,Â u+v)-construction [i] based on
1. digital (11, 18, 6678)-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 (1, 4, 100)-net over F₈₁, using
      - net from sequence [i] based on digital (1, 99)-sequence over F₈₁, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 1 and N(F) ≥ 100, using
        a function field by SÃ©mirat [i]
    81. digital (1, 8, 100)-net over F₈₁, using
      - net from sequence [i] based on digital (1, 99)-sequence over F₈₁ (see above)
2. digital (39, 53, 1198371)-net over F₈₁, using
  - net defined by OOA [i] based on linear OOA(81⁵³, 1198371, F₈₁, 14, 14) (dual of [(1198371, 14), 16777141, 15]-NRT-code), using
    - OA 7-folding and stacking [i] based on linear OA(81⁵³, 8388597, F₈₁, 14) (dual of [8388597, 8388544, 15]-code), using
      - discarding factors / shortening the dual code based on linear OA(81⁵³, large, F₈₁, 14) (dual of [large, large−53, 15]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523360Â | 81⁴âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(57, 71, large)-Net over F₈₁ — Digital

Digital (57, 71, large)-net over F₈₁, using

6 times m-reduction [i] based on digital (57, 77, large)-net over F₈₁, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(81⁷⁷, large, F₈₁, 20) (dual of [large, large−77, 21]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 21523360Â | 81⁴âˆ’1, defining interval IÂ =Â [0,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]

(57, 71, large)-Net in Base 81 — Upper bound on s

There is no (57, 71, large)-net in base 81, because

12 times m-reduction [i] would yield (57, 59, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]

Best Known (57, 71, s)-Nets in Base 81

(57, 71, 1205049)-Net over F81 — Constructive and digital

(57, 71, large)-Net over F81 — Digital

(57, 71, large)-Net in Base 81 — Upper bound on s

(57, 71, 1205049)-Net over F₈₁ — Constructive and digital

(57, 71, large)-Net over F₈₁ — Digital