MinT - Best Known (67, 78, s)-Nets in Base 81

Best Known (67, 78, s)-Nets in Base 81

(67, 78, large)-Net over F₈₁ — Constructive and digital

Digital (67, 78, large)-net over F₈₁, using

81¹⁴ times duplication [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]

(67, 78, large)-Net in Base 81 — Upper bound on s

There is no (67, 78, large)-net in base 81, because

9 times m-reduction [i] would yield (67, 69, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]

Best Known (67, 78, s)-Nets in Base 81

(67, 78, large)-Net over F81 — Constructive and digital

(67, 78, large)-Net in Base 81 — Upper bound on s

(67, 78, large)-Net over F₈₁ — Constructive and digital