MinT - Best Known (69, 69+7, s)-Nets in Base 64

Best Known (69, 69+7, s)-Nets in Base 64

(69, 69+7, large)-Net over F₆₄ — Constructive and digital

Digital (69, 76, large)-net over F₆₄, using

64² times duplication [i] based on digital (67, 74, large)-net over F₆₄, using
- t-expansion [i] based on digital (63, 74, large)-net over F₆₄, using
  - generalized (u,Â u+v)-construction [i] based on
    1. digital (0, 0, 131073)-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, 131073)-net over F₆₄ (see above)
    3. digital (0, 0, 131073)-net over F₆₄ (see above)
    4. digital (0, 0, 131073)-net over F₆₄ (see above)
    5. digital (0, 0, 131073)-net over F₆₄ (see above)
    6. digital (0, 0, 131073)-net over F₆₄ (see above)
    7. digital (0, 0, 131073)-net over F₆₄ (see above)
    8. digital (0, 0, 131073)-net over F₆₄ (see above)
    9. digital (0, 0, 131073)-net over F₆₄ (see above)
    10. digital (0, 0, 131073)-net over F₆₄ (see above)
    11. digital (0, 0, 131073)-net over F₆₄ (see above)
    12. digital (0, 0, 131073)-net over F₆₄ (see above)
    13. digital (0, 0, 131073)-net over F₆₄ (see above)
    14. digital (0, 0, 131073)-net over F₆₄ (see above)
    15. digital (0, 0, 131073)-net over F₆₄ (see above)
    16. digital (0, 0, 131073)-net over F₆₄ (see above)
    17. digital (0, 0, 131073)-net over F₆₄ (see above)
    18. digital (0, 0, 131073)-net over F₆₄ (see above)
    19. digital (0, 0, 131073)-net over F₆₄ (see above)
    20. digital (0, 0, 131073)-net over F₆₄ (see above)
    21. digital (0, 0, 131073)-net over F₆₄ (see above)
    22. digital (0, 0, 131073)-net over F₆₄ (see above)
    23. digital (0, 0, 131073)-net over F₆₄ (see above)
    24. digital (0, 0, 131073)-net over F₆₄ (see above)
    25. digital (0, 0, 131073)-net over F₆₄ (see above)
    26. digital (0, 0, 131073)-net over F₆₄ (see above)
    27. digital (0, 0, 131073)-net over F₆₄ (see above)
    28. digital (0, 0, 131073)-net over F₆₄ (see above)
    29. digital (0, 0, 131073)-net over F₆₄ (see above)
    30. digital (0, 0, 131073)-net over F₆₄ (see above)
    31. digital (0, 0, 131073)-net over F₆₄ (see above)
    32. digital (0, 0, 131073)-net over F₆₄ (see above)
    33. digital (0, 0, 131073)-net over F₆₄ (see above)
    34. digital (0, 0, 131073)-net over F₆₄ (see above)
    35. digital (0, 0, 131073)-net over F₆₄ (see above)
    36. digital (0, 0, 131073)-net over F₆₄ (see above)
    37. digital (0, 0, 131073)-net over F₆₄ (see above)
    38. digital (0, 0, 131073)-net over F₆₄ (see above)
    39. digital (0, 0, 131073)-net over F₆₄ (see above)
    40. digital (0, 0, 131073)-net over F₆₄ (see above)
    41. digital (0, 0, 131073)-net over F₆₄ (see above)
    42. digital (0, 0, 131073)-net over F₆₄ (see above)
    43. digital (0, 0, 131073)-net over F₆₄ (see above)
    44. digital (0, 0, 131073)-net over F₆₄ (see above)
    45. digital (0, 0, 131073)-net over F₆₄ (see above)
    46. digital (0, 0, 131073)-net over F₆₄ (see above)
    47. digital (0, 0, 131073)-net over F₆₄ (see above)
    48. digital (0, 0, 131073)-net over F₆₄ (see above)
    49. digital (0, 0, 131073)-net over F₆₄ (see above)
    50. digital (0, 0, 131073)-net over F₆₄ (see above)
    51. digital (0, 0, 131073)-net over F₆₄ (see above)
    52. digital (0, 0, 131073)-net over F₆₄ (see above)
    53. digital (0, 0, 131073)-net over F₆₄ (see above)
    54. digital (0, 1, 131073)-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]
    55. digital (0, 1, 131073)-net over F₆₄ (see above)
    56. digital (0, 1, 131073)-net over F₆₄ (see above)
    57. digital (0, 1, 131073)-net over F₆₄ (see above)
    58. digital (0, 1, 131073)-net over F₆₄ (see above)
    59. digital (0, 1, 131073)-net over F₆₄ (see above)
    60. digital (2, 4, 131073)-net over F₆₄, using
      - s-reduction based on digital (2, 4, 266305)-net over F₆₄, using
        kÂ =Â 2 construction [i]
    61. digital (2, 4, 131073)-net over F₆₄ (see above)
    62. digital (3, 6, 131073)-net over F₆₄, using
      - s-reduction based on digital (3, 6, 270402)-net over F₆₄, using
        net defined by OOA [i] based on linear OOA(64⁶, 270402, F₆₄, 3, 3) (dual of [(270402, 3), 811200, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(64⁶, 270402, F₆₄, 2, 3) (dual of [(270402, 2), 540798, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(64⁶, 270402, F₆₄, 3) (dual of [270402, 270396, 4]-code or 270402-cap in PG(5,64)), using
        large caps in PG(5,Â b) [i]
    63. digital (8, 13, 131073)-net over F₆₄, using
      - net defined by OOA [i] based on linear OOA(64¹³, 131073, F₆₄, 5, 5) (dual of [(131073, 5), 655352, 6]-NRT-code), using
        OOA 2-folding and stacking with additional row [i] based on linear OA(64¹³, 262147, F₆₄, 5) (dual of [262147, 262134, 6]-code), using
        constructionÂ X applied to C_e(4) âŠ‚ C_e(3) [i] based on
        linear OA(64¹³, 262144, F₆₄, 5) (dual of [262144, 262131, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(64¹⁰, 262144, F₆₄, 4) (dual of [262144, 262134, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(64⁰, 3, F₆₄, 0) (dual of [3, 3, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁰, s, F₆₄, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
    64. digital (30, 41, 1677720)-net over F₆₄, using
      - net defined by OOA [i] based on linear OOA(64⁴¹, 1677720, F₆₄, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
        OOA 5-folding and stacking with additional row [i] based on linear OA(64⁴¹, 8388601, F₆₄, 11) (dual of [8388601, 8388560, 12]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁴¹, large, F₆₄, 11) (dual of [large, large−41, 12]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(69, 69+7, large)-Net in Base 64 — Upper bound on s

There is no (69, 76, large)-net in base 64, because

5 times m-reduction [i] would yield (69, 71, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]

Best Known (69, 69+7, s)-Nets in Base 64

(69, 69+7, large)-Net over F64 — Constructive and digital

(69, 69+7, large)-Net in Base 64 — Upper bound on s

(69, 69+7, large)-Net over F₆₄ — Constructive and digital