MinT - Best Known (32, 40, s)-Nets in Base 8

Best Known (32, 40, s)-Nets in Base 8

(32, 40, 8320)-Net over F₈ — Constructive and digital

Digital (32, 40, 8320)-net over F₈, using

trace code for nets [i] based on digital (12, 20, 4160)-net over F₆₄, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (0, 0, 65)-net over F₆₄, using
    - s-reduction based on digital (0, 0, s)-net over F₆₄ with arbitrarily large s, using
      - arbitrary set of 64⁰ points [i]
  2. digital (0, 0, 65)-net over F₆₄ (see above)
  3. digital (0, 0, 65)-net over F₆₄ (see above)
  4. digital (0, 0, 65)-net over F₆₄ (see above)
  5. digital (0, 0, 65)-net over F₆₄ (see above)
  6. digital (0, 0, 65)-net over F₆₄ (see above)
  7. digital (0, 0, 65)-net over F₆₄ (see above)
  8. digital (0, 0, 65)-net over F₆₄ (see above)
  9. digital (0, 0, 65)-net over F₆₄ (see above)
  10. digital (0, 0, 65)-net over F₆₄ (see above)
  11. digital (0, 0, 65)-net over F₆₄ (see above)
  12. digital (0, 0, 65)-net over F₆₄ (see above)
  13. digital (0, 0, 65)-net over F₆₄ (see above)
  14. digital (0, 0, 65)-net over F₆₄ (see above)
  15. digital (0, 0, 65)-net over F₆₄ (see above)
  16. digital (0, 0, 65)-net over F₆₄ (see above)
  17. digital (0, 0, 65)-net over F₆₄ (see above)
  18. digital (0, 0, 65)-net over F₆₄ (see above)
  19. digital (0, 0, 65)-net over F₆₄ (see above)
  20. digital (0, 0, 65)-net over F₆₄ (see above)
  21. digital (0, 0, 65)-net over F₆₄ (see above)
  22. digital (0, 0, 65)-net over F₆₄ (see above)
  23. digital (0, 0, 65)-net over F₆₄ (see above)
  24. digital (0, 0, 65)-net over F₆₄ (see above)
  25. digital (0, 0, 65)-net over F₆₄ (see above)
  26. digital (0, 0, 65)-net over F₆₄ (see above)
  27. digital (0, 0, 65)-net over F₆₄ (see above)
  28. digital (0, 0, 65)-net over F₆₄ (see above)
  29. digital (0, 0, 65)-net over F₆₄ (see above)
  30. digital (0, 0, 65)-net over F₆₄ (see above)
  31. digital (0, 0, 65)-net over F₆₄ (see above)
  32. digital (0, 0, 65)-net over F₆₄ (see above)
  33. digital (0, 0, 65)-net over F₆₄ (see above)
  34. digital (0, 0, 65)-net over F₆₄ (see above)
  35. digital (0, 0, 65)-net over F₆₄ (see above)
  36. digital (0, 0, 65)-net over F₆₄ (see above)
  37. digital (0, 0, 65)-net over F₆₄ (see above)
  38. digital (0, 0, 65)-net over F₆₄ (see above)
  39. digital (0, 0, 65)-net over F₆₄ (see above)
  40. digital (0, 0, 65)-net over F₆₄ (see above)
  41. digital (0, 0, 65)-net over F₆₄ (see above)
  42. digital (0, 0, 65)-net over F₆₄ (see above)
  43. digital (0, 0, 65)-net over F₆₄ (see above)
  44. digital (0, 0, 65)-net over F₆₄ (see above)
  45. digital (0, 0, 65)-net over F₆₄ (see above)
  46. digital (0, 0, 65)-net over F₆₄ (see above)
  47. digital (0, 0, 65)-net over F₆₄ (see above)
  48. digital (0, 0, 65)-net over F₆₄ (see above)
  49. digital (0, 0, 65)-net over F₆₄ (see above)
  50. digital (0, 0, 65)-net over F₆₄ (see above)
  51. digital (0, 0, 65)-net over F₆₄ (see above)
  52. digital (0, 0, 65)-net over F₆₄ (see above)
  53. digital (0, 0, 65)-net over F₆₄ (see above)
  54. digital (0, 0, 65)-net over F₆₄ (see above)
  55. digital (0, 0, 65)-net over F₆₄ (see above)
  56. digital (0, 0, 65)-net over F₆₄ (see above)
  57. digital (0, 1, 65)-net over F₆₄, using
    - s-reduction based on digital (0, 1, s)-net over F₆₄ with arbitrarily large s, using
      - construction for strength kÂ =Â 1 [i]
  58. digital (0, 1, 65)-net over F₆₄ (see above)
  59. digital (0, 1, 65)-net over F₆₄ (see above)
  60. digital (0, 1, 65)-net over F₆₄ (see above)
  61. digital (0, 2, 65)-net over F₆₄, using
    - kÂ =Â 2 construction [i]
  62. digital (0, 2, 65)-net over F₆₄ (see above)
  63. digital (0, 4, 65)-net over F₆₄, using
    - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
        the rational function field F₆₄(x) [i]
      - Niederreiter sequence [i]
  64. digital (0, 8, 65)-net over F₆₄, using
    - net from sequence [i] based on digital (0, 64)-sequence over F₆₄ (see above)

(32, 40, 32769)-Net in Base 8 — Constructive

(32, 40, 32769)-net in base 8, using

base change [i] based on digital (22, 30, 32769)-net over F₁₆, using
- net defined by OOA [i] based on linear OOA(16³⁰, 32769, F₁₆, 8, 8) (dual of [(32769, 8), 262122, 9]-NRT-code), using
  - OA 4-folding and stacking [i] based on linear OA(16³⁰, 131076, F₁₆, 8) (dual of [131076, 131046, 9]-code), using
    - trace code [i] based on linear OA(256¹⁵, 65538, F₂₅₆, 8) (dual of [65538, 65523, 9]-code), using
      - constructionÂ X applied to C_e(7) âŠ‚ C_e(6) [i] based on
        linear OA(256¹⁵, 65536, F₂₅₆, 8) (dual of [65536, 65521, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
        linear OA(256¹³, 65536, F₂₅₆, 7) (dual of [65536, 65523, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(256⁰, 2, F₂₅₆, 0) (dual of [2, 2, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁰, s, F₂₅₆, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(32, 40, 69880)-Net over F₈ — Digital

Digital (32, 40, 69880)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(32, 40, 131076)-Net in Base 8

(32, 40, 131076)-net in base 8, using

base change [i] based on digital (22, 30, 131076)-net over F₁₆, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16³⁰, 131076, F₁₆, 8) (dual of [131076, 131046, 9]-code), using
  - trace code [i] based on linear OA(256¹⁵, 65538, F₂₅₆, 8) (dual of [65538, 65523, 9]-code), using
    - constructionÂ X applied to C_e(7) âŠ‚ C_e(6) [i] based on
      1. linear OA(256¹⁵, 65536, F₂₅₆, 8) (dual of [65536, 65521, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
      2. linear OA(256¹³, 65536, F₂₅₆, 7) (dual of [65536, 65523, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
      3. linear OA(256⁰, 2, F₂₅₆, 0) (dual of [2, 2, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁰, s, F₂₅₆, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(32, 40, large)-Net in Base 8 — Upper bound on s

There is no (32, 40, large)-net in base 8, because

6 times m-reduction [i] would yield (32, 34, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]