MinT - Best Known (52−33, 52, s)-Nets in Base 64

Best Known (52−33, 52, s)-Nets in Base 64

(52−33, 52, 177)-Net over F₆₄ — Constructive and digital

Digital (19, 52, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 52, 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]

(52−33, 52, 288)-Net in Base 64 — Constructive

(19, 52, 288)-net in base 64, using

18 times m-reduction [i] based on (19, 70, 288)-net in base 64, using
- base change [i] based on digital (9, 60, 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]

(52−33, 52, 315)-Net over F₆₄ — Digital

Digital (19, 52, 315)-net over F₆₄, using

net from sequence [i] based on digital (19, 314)-sequence over F₆₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 19 and N(F) ≥ 315, using
  - a curve from the manYPoints database [i]

(52−33, 52, 321)-Net in Base 64

(19, 52, 321)-net in base 64, using

16 times m-reduction [i] based on (19, 68, 321)-net in base 64, using
- base change [i] based on digital (2, 51, 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]

(52−33, 52, 61708)-Net in Base 64 — Upper bound on s

There is no (19, 52, 61709)-net in base 64, because

1 times m-reduction [i] would yield (19, 51, 61709)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 130 400239 949527 698109 345730 898347 189251 169766 997498 520344 928613 094315 500700 302528 509677 799824 > 64⁵¹ [i]