MinT - Best Known (40−11, 40, s)-Nets in Base 27

Best Known (40−11, 40, s)-Nets in Base 27

(40−11, 40, 4316)-Net over F₂₇ — Constructive and digital

Digital (29, 40, 4316)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (4, 9, 379)-net over F₂₇, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 2, 28)-net over F₂₇, using
      - kÂ =Â 2 construction [i]
    2. digital (2, 7, 351)-net over F₂₇, using
      - net defined by OOA [i] based on linear OOA(27⁷, 351, F₂₇, 5, 5) (dual of [(351, 5), 1748, 6]-NRT-code), using
        OOA 2-folding and stacking with additional row [i] based on linear OA(27⁷, 703, F₂₇, 5) (dual of [703, 696, 6]-code), using
        a code by Danev and Olsson [i]
2. digital (20, 31, 3937)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27³¹, 3937, F₂₇, 11, 11) (dual of [(3937, 11), 43276, 12]-NRT-code), using
    - OOA 5-folding and stacking with additional row [i] based on linear OA(27³¹, 19686, F₂₇, 11) (dual of [19686, 19655, 12]-code), using
      - constructionÂ X applied to C_e(10) âŠ‚ C_e(9) [i] based on
        linear OA(27³¹, 19683, F₂₇, 11) (dual of [19683, 19652, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 27³âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
        linear OA(27²⁸, 19683, F₂₇, 10) (dual of [19683, 19655, 11]-code), using an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 27³âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]
        linear OA(27⁰, 3, F₂₇, 0) (dual of [3, 3, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(27⁰, s, F₂₇, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(40−11, 40, 6660)-Net in Base 27 — Constructive

(29, 40, 6660)-net in base 27, using

base change [i] based on digital (19, 30, 6660)-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, 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]
  72. digital (0, 1, 82)-net over F₈₁ (see above)
  73. digital (0, 1, 82)-net over F₈₁ (see above)
  74. digital (0, 1, 82)-net over F₈₁ (see above)
  75. digital (0, 1, 82)-net over F₈₁ (see above)
  76. digital (0, 1, 82)-net over F₈₁ (see above)
  77. digital (0, 2, 82)-net over F₈₁, using
    - kÂ =Â 2 construction [i]
  78. digital (0, 2, 82)-net over F₈₁ (see above)
  79. digital (0, 3, 82)-net over F₈₁, using
    - net from sequence [i] based on digital (0, 81)-sequence over F₈₁, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 0 and N(F) ≥ 82, using
        the rational function field F₈₁(x) [i]
      - Niederreiter sequence [i]
  80. digital (0, 5, 82)-net over F₈₁, using
    - net from sequence [i] based on digital (0, 81)-sequence over F₈₁ (see above)
  81. digital (1, 12, 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]

(40−11, 40, 92572)-Net over F₂₇ — Digital

Digital (29, 40, 92572)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(40−11, 40, large)-Net in Base 27 — Upper bound on s

There is no (29, 40, large)-net in base 27, because

9 times m-reduction [i] would yield (29, 31, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]

Best Known (40−11, 40, s)-Nets in Base 27

(40−11, 40, 4316)-Net over F27 — Constructive and digital

(40−11, 40, 6660)-Net in Base 27 — Constructive

(40−11, 40, 92572)-Net over F27 — Digital

(40−11, 40, large)-Net in Base 27 — Upper bound on s

(40−11, 40, 4316)-Net over F₂₇ — Constructive and digital

(40−11, 40, 92572)-Net over F₂₇ — Digital