MinT - Best Known (48−6, 48, s)-Nets in Base 81

Best Known (48−6, 48, s)-Nets in Base 81

(48−6, 48, large)-Net over F₈₁ — Constructive and digital

Digital (42, 48, large)-net over F₈₁, using

t-expansion [i] based on digital (41, 48, large)-net over F₈₁, using
- 2 times m-reduction [i] based on 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]

(48−6, 48, large)-Net in Base 81 — Upper bound on s

There is no (42, 48, large)-net in base 81, because

4 times m-reduction [i] would yield (42, 44, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]

Best Known (48−6, 48, s)-Nets in Base 81

(48−6, 48, large)-Net over F81 — Constructive and digital

(48−6, 48, large)-Net in Base 81 — Upper bound on s

(48−6, 48, large)-Net over F₈₁ — Constructive and digital