MinT - Best Known (52−10, 52, s)-Nets in Base 64

Best Known (52−10, 52, s)-Nets in Base 64

(52−10, 52, 1810904)-Net over F₆₄ — Constructive and digital

Digital (42, 52, 1810904)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (10, 15, 133184)-net over F₆₄, using
  - generalized (u,Â u+v)-construction [i] based on
    1. digital (0, 0, 2081)-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, 2081)-net over F₆₄ (see above)
    3. digital (0, 0, 2081)-net over F₆₄ (see above)
    4. digital (0, 0, 2081)-net over F₆₄ (see above)
    5. digital (0, 0, 2081)-net over F₆₄ (see above)
    6. digital (0, 0, 2081)-net over F₆₄ (see above)
    7. digital (0, 0, 2081)-net over F₆₄ (see above)
    8. digital (0, 0, 2081)-net over F₆₄ (see above)
    9. digital (0, 0, 2081)-net over F₆₄ (see above)
    10. digital (0, 0, 2081)-net over F₆₄ (see above)
    11. digital (0, 0, 2081)-net over F₆₄ (see above)
    12. digital (0, 0, 2081)-net over F₆₄ (see above)
    13. digital (0, 0, 2081)-net over F₆₄ (see above)
    14. digital (0, 0, 2081)-net over F₆₄ (see above)
    15. digital (0, 0, 2081)-net over F₆₄ (see above)
    16. digital (0, 0, 2081)-net over F₆₄ (see above)
    17. digital (0, 0, 2081)-net over F₆₄ (see above)
    18. digital (0, 0, 2081)-net over F₆₄ (see above)
    19. digital (0, 0, 2081)-net over F₆₄ (see above)
    20. digital (0, 0, 2081)-net over F₆₄ (see above)
    21. digital (0, 0, 2081)-net over F₆₄ (see above)
    22. digital (0, 0, 2081)-net over F₆₄ (see above)
    23. digital (0, 0, 2081)-net over F₆₄ (see above)
    24. digital (0, 0, 2081)-net over F₆₄ (see above)
    25. digital (0, 0, 2081)-net over F₆₄ (see above)
    26. digital (0, 0, 2081)-net over F₆₄ (see above)
    27. digital (0, 0, 2081)-net over F₆₄ (see above)
    28. digital (0, 0, 2081)-net over F₆₄ (see above)
    29. digital (0, 0, 2081)-net over F₆₄ (see above)
    30. digital (0, 0, 2081)-net over F₆₄ (see above)
    31. digital (0, 0, 2081)-net over F₆₄ (see above)
    32. digital (0, 0, 2081)-net over F₆₄ (see above)
    33. digital (0, 0, 2081)-net over F₆₄ (see above)
    34. digital (0, 0, 2081)-net over F₆₄ (see above)
    35. digital (0, 0, 2081)-net over F₆₄ (see above)
    36. digital (0, 0, 2081)-net over F₆₄ (see above)
    37. digital (0, 0, 2081)-net over F₆₄ (see above)
    38. digital (0, 0, 2081)-net over F₆₄ (see above)
    39. digital (0, 0, 2081)-net over F₆₄ (see above)
    40. digital (0, 0, 2081)-net over F₆₄ (see above)
    41. digital (0, 0, 2081)-net over F₆₄ (see above)
    42. digital (0, 0, 2081)-net over F₆₄ (see above)
    43. digital (0, 0, 2081)-net over F₆₄ (see above)
    44. digital (0, 0, 2081)-net over F₆₄ (see above)
    45. digital (0, 0, 2081)-net over F₆₄ (see above)
    46. digital (0, 0, 2081)-net over F₆₄ (see above)
    47. digital (0, 0, 2081)-net over F₆₄ (see above)
    48. digital (0, 0, 2081)-net over F₆₄ (see above)
    49. digital (0, 0, 2081)-net over F₆₄ (see above)
    50. digital (0, 0, 2081)-net over F₆₄ (see above)
    51. digital (0, 0, 2081)-net over F₆₄ (see above)
    52. digital (0, 0, 2081)-net over F₆₄ (see above)
    53. digital (0, 0, 2081)-net over F₆₄ (see above)
    54. digital (0, 0, 2081)-net over F₆₄ (see above)
    55. digital (0, 0, 2081)-net over F₆₄ (see above)
    56. digital (0, 0, 2081)-net over F₆₄ (see above)
    57. digital (0, 0, 2081)-net over F₆₄ (see above)
    58. digital (0, 0, 2081)-net over F₆₄ (see above)
    59. digital (0, 0, 2081)-net over F₆₄ (see above)
    60. digital (0, 1, 2081)-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]
    61. digital (0, 1, 2081)-net over F₆₄ (see above)
    62. digital (0, 1, 2081)-net over F₆₄ (see above)
    63. digital (1, 3, 2081)-net over F₆₄, using
      - s-reduction based on digital (1, 3, 4161)-net over F₆₄, using
        kÂ =Â 2 construction [i]
    64. digital (4, 9, 2081)-net over F₆₄, using
      - (u,Â u+v)-construction [i] based on
        digital (0, 2, 65)-net over F₆₄, using
        kÂ =Â 2 construction [i]
        digital (2, 7, 2016)-net over F₆₄, using
        net defined by OOA [i] based on linear OOA(64⁷, 2016, F₆₄, 5, 5) (dual of [(2016, 5), 10073, 6]-NRT-code), using
        OOA 2-folding and stacking with additional row [i] based on linear OA(64⁷, 4033, F₆₄, 5) (dual of [4033, 4026, 6]-code), using
        a code by Danev and Olsson [i]
2. digital (27, 37, 1677720)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64³⁷, 1677720, F₆₄, 10, 10) (dual of [(1677720, 10), 16777163, 11]-NRT-code), using
    - OA 5-folding and stacking [i] based on linear OA(64³⁷, 8388600, F₆₄, 10) (dual of [8388600, 8388563, 11]-code), using
      - discarding factors / shortening the dual code based on linear OA(64³⁷, large, F₆₄, 10) (dual of [large, large−37, 11]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(52−10, 52, large)-Net over F₆₄ — Digital

Digital (42, 52, large)-net over F₆₄, using

t-expansion [i] based on digital (41, 52, large)-net over F₆₄, using
- 3 times m-reduction [i] based on digital (41, 55, large)-net over F₆₄, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64⁵⁵, large, F₆₄, 14) (dual of [large, large−55, 15]-code), using
    - 2 times code embedding in larger space [i] based on linear OA(64⁵³, large, F₆₄, 14) (dual of [large, large−53, 15]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(52−10, 52, large)-Net in Base 64 — Upper bound on s

There is no (42, 52, large)-net in base 64, because

8 times m-reduction [i] would yield (42, 44, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]

Best Known (52−10, 52, s)-Nets in Base 64

(52−10, 52, 1810904)-Net over F64 — Constructive and digital

(52−10, 52, large)-Net over F64 — Digital

(52−10, 52, large)-Net in Base 64 — Upper bound on s

(52−10, 52, 1810904)-Net over F₆₄ — Constructive and digital

(52−10, 52, large)-Net over F₆₄ — Digital