MinT - Best Known (56−39, 56, s)-Nets in Base 64

Best Known (56−39, 56, s)-Nets in Base 64

(56−39, 56, 177)-Net over F₆₄ — Constructive and digital

Digital (17, 56, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 56, 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]

(56−39, 56, 267)-Net over F₆₄ — Digital

Digital (17, 56, 267)-net over F₆₄, using

t-expansion [i] based on digital (16, 56, 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]

(56−39, 56, 288)-Net in Base 64 — Constructive

(17, 56, 288)-net in base 64, using

base change [i] based on digital (9, 48, 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]

(56−39, 56, 321)-Net in Base 64

(17, 56, 321)-net in base 64, using

4 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]

(56−39, 56, 21286)-Net in Base 64 — Upper bound on s

There is no (17, 56, 21287)-net in base 64, because

1 times m-reduction [i] would yield (17, 55, 21287)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 2188 542957 438341 850307 909807 609161 386211 235520 930200 695676 256583 483974 367110 876912 538236 805901 308040 > 64⁵⁵ [i]