MinT - Best Known (68−8, 68, s)-Nets in Base 81

Best Known (68−8, 68, s)-Nets in Base 81

(68−8, 68, large)-Net over F₈₁ — Constructive and digital

Digital (60, 68, large)-net over F₈₁, using

81⁴ times duplication [i] based on digital (56, 64, large)-net over F₈₁, using
- t-expansion [i] based on digital (53, 64, large)-net over F₈₁, using
  - generalized (u,Â u+v)-construction [i] based on
    1. digital (0, 0, 106288)-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, 106288)-net over F₈₁ (see above)
    3. digital (0, 0, 106288)-net over F₈₁ (see above)
    4. digital (0, 0, 106288)-net over F₈₁ (see above)
    5. digital (0, 0, 106288)-net over F₈₁ (see above)
    6. digital (0, 0, 106288)-net over F₈₁ (see above)
    7. digital (0, 0, 106288)-net over F₈₁ (see above)
    8. digital (0, 0, 106288)-net over F₈₁ (see above)
    9. digital (0, 0, 106288)-net over F₈₁ (see above)
    10. digital (0, 0, 106288)-net over F₈₁ (see above)
    11. digital (0, 0, 106288)-net over F₈₁ (see above)
    12. digital (0, 0, 106288)-net over F₈₁ (see above)
    13. digital (0, 0, 106288)-net over F₈₁ (see above)
    14. digital (0, 0, 106288)-net over F₈₁ (see above)
    15. digital (0, 0, 106288)-net over F₈₁ (see above)
    16. digital (0, 0, 106288)-net over F₈₁ (see above)
    17. digital (0, 0, 106288)-net over F₈₁ (see above)
    18. digital (0, 0, 106288)-net over F₈₁ (see above)
    19. digital (0, 0, 106288)-net over F₈₁ (see above)
    20. digital (0, 0, 106288)-net over F₈₁ (see above)
    21. digital (0, 0, 106288)-net over F₈₁ (see above)
    22. digital (0, 0, 106288)-net over F₈₁ (see above)
    23. digital (0, 0, 106288)-net over F₈₁ (see above)
    24. digital (0, 0, 106288)-net over F₈₁ (see above)
    25. digital (0, 0, 106288)-net over F₈₁ (see above)
    26. digital (0, 0, 106288)-net over F₈₁ (see above)
    27. digital (0, 0, 106288)-net over F₈₁ (see above)
    28. digital (0, 0, 106288)-net over F₈₁ (see above)
    29. digital (0, 0, 106288)-net over F₈₁ (see above)
    30. digital (0, 0, 106288)-net over F₈₁ (see above)
    31. digital (0, 0, 106288)-net over F₈₁ (see above)
    32. digital (0, 0, 106288)-net over F₈₁ (see above)
    33. digital (0, 0, 106288)-net over F₈₁ (see above)
    34. digital (0, 0, 106288)-net over F₈₁ (see above)
    35. digital (0, 0, 106288)-net over F₈₁ (see above)
    36. digital (0, 0, 106288)-net over F₈₁ (see above)
    37. digital (0, 0, 106288)-net over F₈₁ (see above)
    38. digital (0, 0, 106288)-net over F₈₁ (see above)
    39. digital (0, 0, 106288)-net over F₈₁ (see above)
    40. digital (0, 0, 106288)-net over F₈₁ (see above)
    41. digital (0, 0, 106288)-net over F₈₁ (see above)
    42. digital (0, 0, 106288)-net over F₈₁ (see above)
    43. digital (0, 0, 106288)-net over F₈₁ (see above)
    44. digital (0, 0, 106288)-net over F₈₁ (see above)
    45. digital (0, 0, 106288)-net over F₈₁ (see above)
    46. digital (0, 0, 106288)-net over F₈₁ (see above)
    47. digital (0, 0, 106288)-net over F₈₁ (see above)
    48. digital (0, 0, 106288)-net over F₈₁ (see above)
    49. digital (0, 0, 106288)-net over F₈₁ (see above)
    50. digital (0, 0, 106288)-net over F₈₁ (see above)
    51. digital (0, 0, 106288)-net over F₈₁ (see above)
    52. digital (0, 0, 106288)-net over F₈₁ (see above)
    53. digital (0, 0, 106288)-net over F₈₁ (see above)
    54. digital (0, 0, 106288)-net over F₈₁ (see above)
    55. digital (0, 0, 106288)-net over F₈₁ (see above)
    56. digital (0, 0, 106288)-net over F₈₁ (see above)
    57. digital (0, 0, 106288)-net over F₈₁ (see above)
    58. digital (0, 0, 106288)-net over F₈₁ (see above)
    59. digital (0, 0, 106288)-net over F₈₁ (see above)
    60. digital (0, 0, 106288)-net over F₈₁ (see above)
    61. digital (0, 0, 106288)-net over F₈₁ (see above)
    62. digital (0, 0, 106288)-net over F₈₁ (see above)
    63. digital (0, 0, 106288)-net over F₈₁ (see above)
    64. digital (0, 0, 106288)-net over F₈₁ (see above)
    65. digital (0, 0, 106288)-net over F₈₁ (see above)
    66. digital (0, 0, 106288)-net over F₈₁ (see above)
    67. digital (0, 0, 106288)-net over F₈₁ (see above)
    68. digital (0, 0, 106288)-net over F₈₁ (see above)
    69. digital (0, 0, 106288)-net over F₈₁ (see above)
    70. digital (0, 0, 106288)-net over F₈₁ (see above)
    71. digital (0, 1, 106288)-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]
    72. digital (0, 1, 106288)-net over F₈₁ (see above)
    73. digital (0, 1, 106288)-net over F₈₁ (see above)
    74. digital (0, 1, 106288)-net over F₈₁ (see above)
    75. digital (0, 1, 106288)-net over F₈₁ (see above)
    76. digital (0, 1, 106288)-net over F₈₁ (see above)
    77. digital (2, 4, 106288)-net over F₈₁, using
      - s-reduction based on digital (2, 4, 538084)-net over F₈₁, using
        kÂ =Â 2 construction [i]
    78. digital (2, 4, 106288)-net over F₈₁ (see above)
    79. digital (3, 6, 106288)-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 (8, 13, 106288)-net over F₈₁, using
      - s-reduction based on digital (8, 13, 265721)-net over F₈₁, using
        net defined by OOA [i] based on linear OOA(81¹³, 265721, F₈₁, 5, 5) (dual of [(265721, 5), 1328592, 6]-NRT-code), using
        appending kth column [i] based on linear OOA(81¹³, 265721, F₈₁, 4, 5) (dual of [(265721, 4), 1062871, 6]-NRT-code), using
        OOA 2-folding and stacking with additional row [i] based on linear OA(81¹³, 531443, F₈₁, 5) (dual of [531443, 531430, 6]-code), using
        discarding factors / shortening the dual code based on linear OA(81¹³, 531444, F₈₁, 5) (dual of [531444, 531431, 6]-code), using
        constructionÂ X applied to C_e(4) âŠ‚ C_e(3) [i] based on
        linear OA(81¹³, 531441, F₈₁, 5) (dual of [531441, 531428, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        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(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 (20, 31, 106288)-net over F₈₁, using
      - net defined by OOA [i] based on linear OOA(81³¹, 106288, F₈₁, 11, 11) (dual of [(106288, 11), 1169137, 12]-NRT-code), using
        OOA 5-folding and stacking with additional row [i] based on linear OA(81³¹, 531441, F₈₁, 11) (dual of [531441, 531410, 12]-code), using
        an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(68−8, 68, large)-Net in Base 81 — Upper bound on s

There is no (60, 68, large)-net in base 81, because

6 times m-reduction [i] would yield (60, 62, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]

Best Known (68−8, 68, s)-Nets in Base 81

(68−8, 68, large)-Net over F81 — Constructive and digital

(68−8, 68, large)-Net in Base 81 — Upper bound on s

(68−8, 68, large)-Net over F₈₁ — Constructive and digital