MinT - Best Known (57−40, 57, s)-Nets in Base 64

Best Known (57−40, 57, s)-Nets in Base 64

(57−40, 57, 177)-Net over F₆₄ — Constructive and digital

Digital (17, 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]

(57−40, 57, 259)-Net in Base 64 — Constructive

(17, 57, 259)-net in base 64, using

3 times m-reduction [i] based on (17, 60, 259)-net in base 64, using
- base change [i] based on digital (2, 45, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(57−40, 57, 267)-Net over F₆₄ — Digital

Digital (17, 57, 267)-net over F₆₄, using

t-expansion [i] based on digital (16, 57, 267)-net over F₆₄, using
- net from sequence [i] based on digital (16, 266)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 16 and N(F) ≥ 267, using
    - a curve from the manYPoints database [i]

(57−40, 57, 321)-Net in Base 64

(17, 57, 321)-net in base 64, using

3 times m-reduction [i] based on (17, 60, 321)-net in base 64, using
- base change [i] based on digital (2, 45, 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]

(57−40, 57, 18507)-Net in Base 64 — Upper bound on s

There is no (17, 57, 18508)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 8 963587 201903 080795 178861 506555 505324 833404 652396 839937 744397 248526 138802 784993 252647 367331 648867 131544 > 64⁵⁷ [i]