MinT - Best Known (73−15, 73, s)-Nets in Base 64

Best Known (73−15, 73, s)-Nets in Base 64

(73−15, 73, 1202531)-Net over F₆₄ — Constructive and digital

Digital (58, 73, 1202531)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (9, 16, 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, 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]
    59. digital (0, 1, 65)-net over F₆₄ (see above)
    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, 7, 65)-net over F₆₄, using
      - net from sequence [i] based on digital (0, 64)-sequence over F₆₄ (see above)
2. digital (42, 57, 1198371)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64⁵⁷, 1198371, F₆₄, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
    - OOA 7-folding and stacking with additional row [i] based on linear OA(64⁵⁷, 8388598, F₆₄, 15) (dual of [8388598, 8388541, 16]-code), using
      - discarding factors / shortening the dual code 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]

(73−15, 73, 1203832)-Net in Base 64 — Constructive

(58, 73, 1203832)-net in base 64, using

(u,Â u+v)-construction [i] based on
1. (9, 16, 5461)-net in base 64, using
  - net defined by OOA [i] based on OOA(64¹⁶, 5461, S₆₄, 7, 7), using
    - OOA 3-folding and stacking with additional row [i] based on OA(64¹⁶, 16384, S₆₄, 7), using
      - discarding factors based on OA(64¹⁶, 16386, S₆₄, 7), using
        discarding parts of the base [i] based on linear OA(128¹³, 16386, F₁₂₈, 7) (dual of [16386, 16373, 8]-code), using
        constructionÂ X applied to C_e(6) âŠ‚ C_e(5) [i] based on
        linear OA(128¹³, 16384, F₁₂₈, 7) (dual of [16384, 16371, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 128²âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        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(128⁰, 2, F₁₂₈, 0) (dual of [2, 2, 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 (42, 57, 1198371)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64⁵⁷, 1198371, F₆₄, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
    - OOA 7-folding and stacking with additional row [i] based on linear OA(64⁵⁷, 8388598, F₆₄, 15) (dual of [8388598, 8388541, 16]-code), using
      - discarding factors / shortening the dual code 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]

(73−15, 73, large)-Net over F₆₄ — Digital

Digital (58, 73, large)-net over F₆₄, using

t-expansion [i] based on digital (57, 73, large)-net over F₆₄, using
- 3 times m-reduction [i] based on digital (57, 76, large)-net over F₆₄, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64⁷⁶, large, F₆₄, 19) (dual of [large, large−76, 20]-code), using
    - 3 times code embedding in larger space [i] based on linear OA(64⁷³, large, F₆₄, 19) (dual of [large, large−73, 20]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]

(73−15, 73, large)-Net in Base 64 — Upper bound on s

There is no (58, 73, large)-net in base 64, because

13 times m-reduction [i] would yield (58, 60, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]

Best Known (73−15, 73, s)-Nets in Base 64

(73−15, 73, 1202531)-Net over F64 — Constructive and digital

(73−15, 73, 1203832)-Net in Base 64 — Constructive

(73−15, 73, large)-Net over F64 — Digital

(73−15, 73, large)-Net in Base 64 — Upper bound on s

(73−15, 73, 1202531)-Net over F₆₄ — Constructive and digital

(73−15, 73, large)-Net over F₆₄ — Digital