MinT - Best Known (57−39, 57, s)-Nets in Base 64

Best Known (57−39, 57, s)-Nets in Base 64

(57−39, 57, 177)-Net over F₆₄ — Constructive and digital

Digital (18, 57, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 57, 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]

(57−39, 57, 281)-Net over F₆₄ — Digital

Digital (18, 57, 281)-net over F₆₄, using

net from sequence [i] based on digital (18, 280)-sequence over F₆₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 18 and N(F) ≥ 281, using
  - a curve from the manYPoints database [i]

(57−39, 57, 288)-Net in Base 64 — Constructive

(18, 57, 288)-net in base 64, using

6 times m-reduction [i] based on (18, 63, 288)-net in base 64, using
- base change [i] based on digital (9, 54, 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]

(57−39, 57, 321)-Net in Base 64

(18, 57, 321)-net in base 64, using

7 times m-reduction [i] based on (18, 64, 321)-net in base 64, using
- base change [i] based on digital (2, 48, 321)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 320)-sequence over F₂₅₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 2 and N(F) ≥ 321, using
      - sharp bound on the number of rational points on curves with genus 2 [i]

(57−39, 57, 26497)-Net in Base 64 — Upper bound on s

There is no (18, 57, 26498)-net in base 64, because

1 times m-reduction [i] would yield (18, 56, 26498)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 140062 049400 561604 265733 062289 992257 481673 667093 133038 472214 361936 548421 111348 260967 843917 376319 400256 > 64⁵⁶ [i]