MinT - Best Known (69, 69+12, s)-Nets in Base 81

Best Known (69, 69+12, s)-Nets in Base 81

(69, 69+12, 7174494)-Net over F₈₁ — Constructive and digital

Digital (69, 81, 7174494)-net over F₈₁, using

81¹ times duplication [i] based on digital (68, 80, 7174494)-net over F₈₁, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (0, 0, 88574)-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, 88574)-net over F₈₁ (see above)
  3. digital (0, 0, 88574)-net over F₈₁ (see above)
  4. digital (0, 0, 88574)-net over F₈₁ (see above)
  5. digital (0, 0, 88574)-net over F₈₁ (see above)
  6. digital (0, 0, 88574)-net over F₈₁ (see above)
  7. digital (0, 0, 88574)-net over F₈₁ (see above)
  8. digital (0, 0, 88574)-net over F₈₁ (see above)
  9. digital (0, 0, 88574)-net over F₈₁ (see above)
  10. digital (0, 0, 88574)-net over F₈₁ (see above)
  11. digital (0, 0, 88574)-net over F₈₁ (see above)
  12. digital (0, 0, 88574)-net over F₈₁ (see above)
  13. digital (0, 0, 88574)-net over F₈₁ (see above)
  14. digital (0, 0, 88574)-net over F₈₁ (see above)
  15. digital (0, 0, 88574)-net over F₈₁ (see above)
  16. digital (0, 0, 88574)-net over F₈₁ (see above)
  17. digital (0, 0, 88574)-net over F₈₁ (see above)
  18. digital (0, 0, 88574)-net over F₈₁ (see above)
  19. digital (0, 0, 88574)-net over F₈₁ (see above)
  20. digital (0, 0, 88574)-net over F₈₁ (see above)
  21. digital (0, 0, 88574)-net over F₈₁ (see above)
  22. digital (0, 0, 88574)-net over F₈₁ (see above)
  23. digital (0, 0, 88574)-net over F₈₁ (see above)
  24. digital (0, 0, 88574)-net over F₈₁ (see above)
  25. digital (0, 0, 88574)-net over F₈₁ (see above)
  26. digital (0, 0, 88574)-net over F₈₁ (see above)
  27. digital (0, 0, 88574)-net over F₈₁ (see above)
  28. digital (0, 0, 88574)-net over F₈₁ (see above)
  29. digital (0, 0, 88574)-net over F₈₁ (see above)
  30. digital (0, 0, 88574)-net over F₈₁ (see above)
  31. digital (0, 0, 88574)-net over F₈₁ (see above)
  32. digital (0, 0, 88574)-net over F₈₁ (see above)
  33. digital (0, 0, 88574)-net over F₈₁ (see above)
  34. digital (0, 0, 88574)-net over F₈₁ (see above)
  35. digital (0, 0, 88574)-net over F₈₁ (see above)
  36. digital (0, 0, 88574)-net over F₈₁ (see above)
  37. digital (0, 0, 88574)-net over F₈₁ (see above)
  38. digital (0, 0, 88574)-net over F₈₁ (see above)
  39. digital (0, 0, 88574)-net over F₈₁ (see above)
  40. digital (0, 0, 88574)-net over F₈₁ (see above)
  41. digital (0, 0, 88574)-net over F₈₁ (see above)
  42. digital (0, 0, 88574)-net over F₈₁ (see above)
  43. digital (0, 0, 88574)-net over F₈₁ (see above)
  44. digital (0, 0, 88574)-net over F₈₁ (see above)
  45. digital (0, 0, 88574)-net over F₈₁ (see above)
  46. digital (0, 0, 88574)-net over F₈₁ (see above)
  47. digital (0, 0, 88574)-net over F₈₁ (see above)
  48. digital (0, 0, 88574)-net over F₈₁ (see above)
  49. digital (0, 0, 88574)-net over F₈₁ (see above)
  50. digital (0, 0, 88574)-net over F₈₁ (see above)
  51. digital (0, 0, 88574)-net over F₈₁ (see above)
  52. digital (0, 0, 88574)-net over F₈₁ (see above)
  53. digital (0, 0, 88574)-net over F₈₁ (see above)
  54. digital (0, 0, 88574)-net over F₈₁ (see above)
  55. digital (0, 0, 88574)-net over F₈₁ (see above)
  56. digital (0, 0, 88574)-net over F₈₁ (see above)
  57. digital (0, 0, 88574)-net over F₈₁ (see above)
  58. digital (0, 0, 88574)-net over F₈₁ (see above)
  59. digital (0, 0, 88574)-net over F₈₁ (see above)
  60. digital (0, 0, 88574)-net over F₈₁ (see above)
  61. digital (0, 0, 88574)-net over F₈₁ (see above)
  62. digital (0, 0, 88574)-net over F₈₁ (see above)
  63. digital (0, 0, 88574)-net over F₈₁ (see above)
  64. digital (0, 0, 88574)-net over F₈₁ (see above)
  65. digital (0, 0, 88574)-net over F₈₁ (see above)
  66. digital (0, 0, 88574)-net over F₈₁ (see above)
  67. digital (0, 0, 88574)-net over F₈₁ (see above)
  68. digital (0, 0, 88574)-net over F₈₁ (see above)
  69. digital (0, 0, 88574)-net over F₈₁ (see above)
  70. digital (0, 1, 88574)-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]
  71. digital (0, 1, 88574)-net over F₈₁ (see above)
  72. digital (0, 1, 88574)-net over F₈₁ (see above)
  73. digital (0, 1, 88574)-net over F₈₁ (see above)
  74. digital (0, 1, 88574)-net over F₈₁ (see above)
  75. digital (0, 1, 88574)-net over F₈₁ (see above)
  76. digital (2, 4, 88574)-net over F₈₁, using
    - s-reduction based on digital (2, 4, 538084)-net over F₈₁, using
      - kÂ =Â 2 construction [i]
  77. digital (2, 4, 88574)-net over F₈₁ (see above)
  78. digital (3, 6, 88574)-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]
  79. digital (6, 10, 88574)-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]
  80. digital (10, 16, 88574)-net over F₈₁, using
    - s-reduction based on digital (10, 16, 177148)-net over F₈₁, using
      - net defined by OOA [i] based on linear OOA(81¹⁶, 177148, F₈₁, 6, 6) (dual of [(177148, 6), 1062872, 7]-NRT-code), using
        OA 3-folding and stacking [i] based on linear OA(81¹⁶, 531444, F₈₁, 6) (dual of [531444, 531428, 7]-code), using
        constructionÂ X applied to C_e(5) âŠ‚ C_e(4) [i] based on
        linear OA(81¹⁶, 531441, F₈₁, 6) (dual of [531441, 531425, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        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(81⁰, 3, F₈₁, 0) (dual of [3, 3, 1]-code) (see above)
  81. digital (22, 34, 88574)-net over F₈₁, using
    - net defined by OOA [i] based on linear OOA(81³⁴, 88574, F₈₁, 12, 12) (dual of [(88574, 12), 1062854, 13]-NRT-code), using
      - OA 6-folding and stacking [i] based on linear OA(81³⁴, 531444, F₈₁, 12) (dual of [531444, 531410, 13]-code), using
        constructionÂ X applied to C_e(11) âŠ‚ C_e(10) [i] based on
        linear OA(81³⁴, 531441, F₈₁, 12) (dual of [531441, 531407, 13]-code), using an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
        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]
        linear OA(81⁰, 3, F₈₁, 0) (dual of [3, 3, 1]-code) (see above)

(69, 69+12, large)-Net over F₈₁ — Digital

Digital (69, 81, large)-net over F₈₁, using

t-expansion [i] based on digital (60, 81, large)-net over F₈₁, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(81⁸¹, large, F₈₁, 21) (dual of [large, large−81, 22]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 21523361Â | 81⁸âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]

(69, 69+12, large)-Net in Base 81 — Upper bound on s

There is no (69, 81, large)-net in base 81, because

10 times m-reduction [i] would yield (69, 71, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]

Best Known (69, 69+12, s)-Nets in Base 81

(69, 69+12, 7174494)-Net over F81 — Constructive and digital

(69, 69+12, large)-Net over F81 — Digital

(69, 69+12, large)-Net in Base 81 — Upper bound on s

(69, 69+12, 7174494)-Net over F₈₁ — Constructive and digital

(69, 69+12, large)-Net over F₈₁ — Digital