MinT - Best Known (48, 48+6, s)-Nets in Base 64

Best Known (48, 48+6, s)-Nets in Base 64

(48, 48+6, large)-Net over F₆₄ — Constructive and digital

Digital (48, 54, large)-net over F₆₄, using

64³ times duplication [i] based on digital (45, 51, large)-net over F₆₄, using
- t-expansion [i] based on digital (43, 51, 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, 0, 131073)-net over F₆₄ (see above)
    55. digital (0, 0, 131073)-net over F₆₄ (see above)
    56. digital (0, 0, 131073)-net over F₆₄ (see above)
    57. 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]
    58. digital (0, 1, 131073)-net over F₆₄ (see above)
    59. digital (0, 1, 131073)-net over F₆₄ (see above)
    60. digital (0, 1, 131073)-net over F₆₄ (see above)
    61. 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]
    62. digital (2, 4, 131073)-net over F₆₄ (see above)
    63. digital (6, 10, 131073)-net over F₆₄, using
      - net defined by OOA [i] based on linear OOA(64¹⁰, 131073, F₆₄, 4, 4) (dual of [(131073, 4), 524282, 5]-NRT-code), using
        appending kth column [i] based on linear OOA(64¹⁰, 131073, F₆₄, 3, 4) (dual of [(131073, 3), 393209, 5]-NRT-code), using
        OA 2-folding and stacking [i] based on linear OA(64¹⁰, 262146, F₆₄, 4) (dual of [262146, 262136, 5]-code), using
        discarding factors / shortening the dual code based on linear OA(64¹⁰, 262147, F₆₄, 4) (dual of [262147, 262137, 5]-code), using
        constructionÂ X applied to C_e(3) âŠ‚ C_e(2) [i] based on
        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(64⁷, 262144, F₆₄, 3) (dual of [262144, 262137, 4]-code or 262144-cap in PG(6,64)), using an extension C_e(2) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [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 (21, 29, 2097150)-net over F₆₄, using
      - net defined by OOA [i] based on linear OOA(64²⁹, 2097150, F₆₄, 8, 8) (dual of [(2097150, 8), 16777171, 9]-NRT-code), using
        OA 4-folding and stacking [i] based on linear OA(64²⁹, 8388600, F₆₄, 8) (dual of [8388600, 8388571, 9]-code), using
        discarding factors / shortening the dual code based on linear OA(64²⁹, large, F₆₄, 8) (dual of [large, large−29, 9]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]

(48, 48+6, large)-Net in Base 64 — Upper bound on s

There is no (48, 54, large)-net in base 64, because

4 times m-reduction [i] would yield (48, 50, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]

Best Known (48, 48+6, s)-Nets in Base 64

(48, 48+6, large)-Net over F64 — Constructive and digital

(48, 48+6, large)-Net in Base 64 — Upper bound on s

(48, 48+6, large)-Net over F₆₄ — Constructive and digital