MinT - Best Known (68, 85, s)-Nets in Base 64

Best Known (68, 85, s)-Nets in Base 64

(68, 85, 1052735)-Net over F₆₄ — Constructive and digital

Digital (68, 85, 1052735)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. 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)
2. digital (48, 65, 1048575)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64⁶⁵, 1048575, F₆₄, 17, 17) (dual of [(1048575, 17), 17825710, 18]-NRT-code), using
    - OOA 8-folding and stacking with additional row [i] based on linear OA(64⁶⁵, 8388601, F₆₄, 17) (dual of [8388601, 8388536, 18]-code), using
      - discarding factors / shortening the dual code based on linear OA(64⁶⁵, large, F₆₄, 17) (dual of [large, large−65, 18]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(68, 85, 1064959)-Net in Base 64 — Constructive

(68, 85, 1064959)-net in base 64, using

(u,Â u+v)-construction [i] based on
1. (12, 20, 16384)-net in base 64, using
  - base change [i] based on digital (7, 15, 16384)-net over F₂₅₆, using
    - net defined by OOA [i] based on linear OOA(256¹⁵, 16384, F₂₅₆, 8, 8) (dual of [(16384, 8), 131057, 9]-NRT-code), using
      - OA 4-folding and stacking [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]
2. digital (48, 65, 1048575)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64⁶⁵, 1048575, F₆₄, 17, 17) (dual of [(1048575, 17), 17825710, 18]-NRT-code), using
    - OOA 8-folding and stacking with additional row [i] based on linear OA(64⁶⁵, 8388601, F₆₄, 17) (dual of [8388601, 8388536, 18]-code), using
      - discarding factors / shortening the dual code based on linear OA(64⁶⁵, large, F₆₄, 17) (dual of [large, large−65, 18]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(68, 85, large)-Net over F₆₄ — Digital

Digital (68, 85, large)-net over F₆₄, using

t-expansion [i] based on digital (66, 85, large)-net over F₆₄, using
- 3 times m-reduction [i] based on digital (66, 88, large)-net over F₆₄, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64⁸⁸, large, F₆₄, 22) (dual of [large, large−88, 23]-code), using
    - 3 times code embedding in larger space [i] based on linear OA(64⁸⁵, large, F₆₄, 22) (dual of [large, large−85, 23]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(68, 85, large)-Net in Base 64 — Upper bound on s

There is no (68, 85, large)-net in base 64, because

15 times m-reduction [i] would yield (68, 70, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]

Best Known (68, 85, s)-Nets in Base 64

(68, 85, 1052735)-Net over F64 — Constructive and digital

(68, 85, 1064959)-Net in Base 64 — Constructive

(68, 85, large)-Net over F64 — Digital

(68, 85, large)-Net in Base 64 — Upper bound on s

(68, 85, 1052735)-Net over F₆₄ — Constructive and digital

(68, 85, large)-Net over F₆₄ — Digital