MinT - Best Known (50, 50+13, s)-Nets in Base 64

Best Known (50, 50+13, s)-Nets in Base 64

(50, 50+13, 1402260)-Net over F₆₄ — Constructive and digital

Digital (50, 63, 1402260)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (8, 14, 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, 0, 65)-net over F₆₄ (see above)
    58. digital (0, 0, 65)-net over F₆₄ (see above)
    59. 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]
    60. digital (0, 1, 65)-net over F₆₄ (see above)
    61. digital (0, 1, 65)-net over F₆₄ (see above)
    62. digital (0, 2, 65)-net over F₆₄, using
      - kÂ =Â 2 construction [i]
    63. digital (0, 3, 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, 6, 65)-net over F₆₄, using
      - net from sequence [i] based on digital (0, 64)-sequence over F₆₄ (see above)
2. digital (36, 49, 1398100)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64⁴⁹, 1398100, F₆₄, 13, 13) (dual of [(1398100, 13), 18175251, 14]-NRT-code), using
    - OOA 6-folding and stacking with additional row [i] based on linear OA(64⁴⁹, 8388601, F₆₄, 13) (dual of [8388601, 8388552, 14]-code), using
      - discarding factors / shortening the dual code based on linear OA(64⁴⁹, large, F₆₄, 13) (dual of [large, large−49, 14]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(50, 50+13, 1403563)-Net in Base 64 — Constructive

(50, 63, 1403563)-net in base 64, using

(u,Â u+v)-construction [i] based on
1. (8, 14, 5463)-net in base 64, using
  - base change [i] based on digital (6, 12, 5463)-net over F₁₂₈, using
    - net defined by OOA [i] based on linear OOA(128¹², 5463, F₁₂₈, 6, 6) (dual of [(5463, 6), 32766, 7]-NRT-code), using
      - OA 3-folding and stacking [i] based on linear OA(128¹², 16389, F₁₂₈, 6) (dual of [16389, 16377, 7]-code), using
        constructionÂ X applied to C_e(5) âŠ‚ C_e(3) [i] based on
        linear OA(128¹¹, 16384, F₁₂₈, 6) (dual of [16384, 16373, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 128²âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(128⁷, 16384, F₁₂₈, 4) (dual of [16384, 16377, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 128²âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(128¹, 5, F₁₂₈, 1) (dual of [5, 4, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(128¹, s, F₁₂₈, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]
2. digital (36, 49, 1398100)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64⁴⁹, 1398100, F₆₄, 13, 13) (dual of [(1398100, 13), 18175251, 14]-NRT-code), using
    - OOA 6-folding and stacking with additional row [i] based on linear OA(64⁴⁹, 8388601, F₆₄, 13) (dual of [8388601, 8388552, 14]-code), using
      - discarding factors / shortening the dual code based on linear OA(64⁴⁹, large, F₆₄, 13) (dual of [large, large−49, 14]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(50, 50+13, large)-Net over F₆₄ — Digital

Digital (50, 63, large)-net over F₆₄, using

4 times m-reduction [i] based on digital (50, 67, large)-net over F₆₄, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64⁶⁷, large, F₆₄, 17) (dual of [large, large−67, 18]-code), using
  - 2 times code embedding in larger space [i] 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]

(50, 50+13, large)-Net in Base 64 — Upper bound on s

There is no (50, 63, large)-net in base 64, because

11 times m-reduction [i] would yield (50, 52, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]

Best Known (50, 50+13, s)-Nets in Base 64

(50, 50+13, 1402260)-Net over F64 — Constructive and digital

(50, 50+13, 1403563)-Net in Base 64 — Constructive

(50, 50+13, large)-Net over F64 — Digital

(50, 50+13, large)-Net in Base 64 — Upper bound on s

(50, 50+13, 1402260)-Net over F₆₄ — Constructive and digital

(50, 50+13, large)-Net over F₆₄ — Digital