MinT - Best Known (59−37, 59, s)-Nets in Base 64

Best Known (59−37, 59, s)-Nets in Base 64

(59−37, 59, 184)-Net over F₆₄ — Constructive and digital

Digital (22, 59, 184)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 19, 80)-net over F₆₄, using
  - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
      - a function field by SÃ©mirat [i]
2. digital (3, 40, 104)-net over F₆₄, using
  - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(59−37, 59, 288)-Net in Base 64 — Constructive

(22, 59, 288)-net in base 64, using

32 times m-reduction [i] based on (22, 91, 288)-net in base 64, using
- base change [i] based on digital (9, 78, 288)-net over F₁₂₈, using
  - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
      - a function field by SÃ©mirat [i]

(59−37, 59, 342)-Net over F₆₄ — Digital

Digital (22, 59, 342)-net over F₆₄, using

t-expansion [i] based on digital (20, 59, 342)-net over F₆₄, using
- net from sequence [i] based on digital (20, 341)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 20 and N(F) ≥ 342, using
    - a curve from the manYPoints database [i]

(59−37, 59, 79187)-Net in Base 64 — Upper bound on s

There is no (22, 59, 79188)-net in base 64, because

1 times m-reduction [i] would yield (22, 58, 79188)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 573 488682 370192 051052 744619 063087 983699 554672 754751 878652 508392 842052 322184 936109 705160 602937 033150 242055 > 64⁵⁸ [i]