MinT - Best Known (57−31, 57, s)-Nets in Base 64

Best Known (57−31, 57, s)-Nets in Base 64

(57−31, 57, 281)-Net over F₆₄ — Constructive and digital

Digital (26, 57, 281)-net over F₆₄, using

1 times m-reduction [i] based on digital (26, 58, 281)-net over F₆₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (3, 19, 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]
  2. digital (7, 39, 177)-net over F₆₄, using
    - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
        a function field by GarcÃa and Quoos [i]

(57−31, 57, 327)-Net in Base 64 — Constructive

(26, 57, 327)-net in base 64, using

(u,Â u+v)-construction [i] based on
1. (4, 19, 150)-net in base 64, using
  - 2 times m-reduction [i] based on (4, 21, 150)-net in base 64, using
    - base change [i] based on digital (1, 18, 150)-net over F₁₂₈, using
      - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
        a function field by SÃ©mirat [i]
2. digital (7, 38, 177)-net over F₆₄, using
  - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
      - a function field by GarcÃa and Quoos [i]

(57−31, 57, 532)-Net over F₆₄ — Digital

Digital (26, 57, 532)-net over F₆₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(57−31, 57, 564277)-Net in Base 64 — Upper bound on s

There is no (26, 57, 564278)-net in base 64, because

1 times m-reduction [i] would yield (26, 56, 564278)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 139987 406388 991827 501216 753943 788667 732233 037492 042862 726077 099295 177944 401286 109173 252812 340508 086264 > 64⁵⁶ [i]