MinT - Best Known (57−11, 57, s)-Nets in Base 64

Best Known (57−11, 57, s)-Nets in Base 64

(57−11, 57, 1943960)-Net over F₆₄ — Constructive and digital

Digital (46, 57, 1943960)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (11, 16, 266240)-net over F₆₄, using
  - generalized (u,Â u+v)-construction [i] based on
    1. digital (0, 0, 4160)-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, 4160)-net over F₆₄ (see above)
    3. digital (0, 0, 4160)-net over F₆₄ (see above)
    4. digital (0, 0, 4160)-net over F₆₄ (see above)
    5. digital (0, 0, 4160)-net over F₆₄ (see above)
    6. digital (0, 0, 4160)-net over F₆₄ (see above)
    7. digital (0, 0, 4160)-net over F₆₄ (see above)
    8. digital (0, 0, 4160)-net over F₆₄ (see above)
    9. digital (0, 0, 4160)-net over F₆₄ (see above)
    10. digital (0, 0, 4160)-net over F₆₄ (see above)
    11. digital (0, 0, 4160)-net over F₆₄ (see above)
    12. digital (0, 0, 4160)-net over F₆₄ (see above)
    13. digital (0, 0, 4160)-net over F₆₄ (see above)
    14. digital (0, 0, 4160)-net over F₆₄ (see above)
    15. digital (0, 0, 4160)-net over F₆₄ (see above)
    16. digital (0, 0, 4160)-net over F₆₄ (see above)
    17. digital (0, 0, 4160)-net over F₆₄ (see above)
    18. digital (0, 0, 4160)-net over F₆₄ (see above)
    19. digital (0, 0, 4160)-net over F₆₄ (see above)
    20. digital (0, 0, 4160)-net over F₆₄ (see above)
    21. digital (0, 0, 4160)-net over F₆₄ (see above)
    22. digital (0, 0, 4160)-net over F₆₄ (see above)
    23. digital (0, 0, 4160)-net over F₆₄ (see above)
    24. digital (0, 0, 4160)-net over F₆₄ (see above)
    25. digital (0, 0, 4160)-net over F₆₄ (see above)
    26. digital (0, 0, 4160)-net over F₆₄ (see above)
    27. digital (0, 0, 4160)-net over F₆₄ (see above)
    28. digital (0, 0, 4160)-net over F₆₄ (see above)
    29. digital (0, 0, 4160)-net over F₆₄ (see above)
    30. digital (0, 0, 4160)-net over F₆₄ (see above)
    31. digital (0, 0, 4160)-net over F₆₄ (see above)
    32. digital (0, 0, 4160)-net over F₆₄ (see above)
    33. digital (0, 0, 4160)-net over F₆₄ (see above)
    34. digital (0, 0, 4160)-net over F₆₄ (see above)
    35. digital (0, 0, 4160)-net over F₆₄ (see above)
    36. digital (0, 0, 4160)-net over F₆₄ (see above)
    37. digital (0, 0, 4160)-net over F₆₄ (see above)
    38. digital (0, 0, 4160)-net over F₆₄ (see above)
    39. digital (0, 0, 4160)-net over F₆₄ (see above)
    40. digital (0, 0, 4160)-net over F₆₄ (see above)
    41. digital (0, 0, 4160)-net over F₆₄ (see above)
    42. digital (0, 0, 4160)-net over F₆₄ (see above)
    43. digital (0, 0, 4160)-net over F₆₄ (see above)
    44. digital (0, 0, 4160)-net over F₆₄ (see above)
    45. digital (0, 0, 4160)-net over F₆₄ (see above)
    46. digital (0, 0, 4160)-net over F₆₄ (see above)
    47. digital (0, 0, 4160)-net over F₆₄ (see above)
    48. digital (0, 0, 4160)-net over F₆₄ (see above)
    49. digital (0, 0, 4160)-net over F₆₄ (see above)
    50. digital (0, 0, 4160)-net over F₆₄ (see above)
    51. digital (0, 0, 4160)-net over F₆₄ (see above)
    52. digital (0, 0, 4160)-net over F₆₄ (see above)
    53. digital (0, 0, 4160)-net over F₆₄ (see above)
    54. digital (0, 0, 4160)-net over F₆₄ (see above)
    55. digital (0, 0, 4160)-net over F₆₄ (see above)
    56. digital (0, 0, 4160)-net over F₆₄ (see above)
    57. digital (0, 0, 4160)-net over F₆₄ (see above)
    58. digital (0, 0, 4160)-net over F₆₄ (see above)
    59. digital (0, 0, 4160)-net over F₆₄ (see above)
    60. digital (0, 1, 4160)-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, 4160)-net over F₆₄ (see above)
    62. digital (0, 1, 4160)-net over F₆₄ (see above)
    63. digital (1, 3, 4160)-net over F₆₄, using
      - s-reduction based on digital (1, 3, 4161)-net over F₆₄, using
        kÂ =Â 2 construction [i]
    64. digital (5, 10, 4160)-net over F₆₄, using
      - generalized (u,Â u+v)-construction [i] based on
        digital (0, 0, 65)-net over F₆₄, using
        s-reduction based on digital (0, 0, s)-net over F₆₄ with arbitrarily large s (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 0, 65)-net over F₆₄ (see above)
        digital (0, 1, 65)-net over F₆₄, using
        s-reduction based on digital (0, 1, s)-net over F₆₄ with arbitrarily large s (see above)
        digital (0, 1, 65)-net over F₆₄ (see above)
        digital (0, 1, 65)-net over F₆₄ (see above)
        digital (0, 2, 65)-net over F₆₄, using
        kÂ =Â 2 construction [i]
        digital (0, 5, 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]
2. 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]

(57−11, 57, 2726297)-Net in Base 64 — Constructive

(46, 57, 2726297)-net in base 64, using

(u,Â u+v)-construction [i] based on
1. (11, 16, 1048577)-net in base 64, using
  - net defined by OOA [i] based on OOA(64¹⁶, 1048577, S₆₄, 5, 5), using
    - OOA 2-folding and stacking with additional row [i] based on OA(64¹⁶, 2097155, S₆₄, 5), using
      - discarding parts of the base [i] based on linear OA(128¹³, 2097155, F₁₂₈, 5) (dual of [2097155, 2097142, 6]-code), using
        constructionÂ X applied to C_e(4) âŠ‚ C_e(3) [i] based on
        linear OA(128¹³, 2097152, F₁₂₈, 5) (dual of [2097152, 2097139, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(128¹⁰, 2097152, F₁₂₈, 4) (dual of [2097152, 2097142, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(128⁰, 3, F₁₂₈, 0) (dual of [3, 3, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(128⁰, s, F₁₂₈, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
2. 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]

(57−11, 57, large)-Net over F₆₄ — Digital

Digital (46, 57, large)-net over F₆₄, using

t-expansion [i] based on digital (44, 57, large)-net over F₆₄, using
- 2 times m-reduction [i] based on digital (44, 59, large)-net over F₆₄, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64⁵⁹, large, F₆₄, 15) (dual of [large, large−59, 16]-code), using
    - 2 times code embedding in larger space [i] based on linear OA(64⁵⁷, large, F₆₄, 15) (dual of [large, large−57, 16]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]

(57−11, 57, large)-Net in Base 64 — Upper bound on s

There is no (46, 57, large)-net in base 64, because

9 times m-reduction [i] would yield (46, 48, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]

Best Known (57−11, 57, s)-Nets in Base 64

(57−11, 57, 1943960)-Net over F64 — Constructive and digital

(57−11, 57, 2726297)-Net in Base 64 — Constructive

(57−11, 57, large)-Net over F64 — Digital

(57−11, 57, large)-Net in Base 64 — Upper bound on s

(57−11, 57, 1943960)-Net over F₆₄ — Constructive and digital

(57−11, 57, large)-Net over F₆₄ — Digital