MinT - Best Known (31−21, 31, s)-Nets in Base 64

Best Known (31−21, 31, s)-Nets in Base 64

(31−21, 31, 177)-Net over F₆₄ — Constructive and digital

Digital (10, 31, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 31, 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]

(31−21, 31, 225)-Net over F₆₄ — Digital

Digital (10, 31, 225)-net over F₆₄, using

net from sequence [i] based on digital (10, 224)-sequence over F₆₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 10 and N(F) ≥ 225, using
  - a curve from the manYPoints database [i]

(31−21, 31, 259)-Net in Base 64 — Constructive

(10, 31, 259)-net in base 64, using

1 times m-reduction [i] based on (10, 32, 259)-net in base 64, using
- base change [i] based on digital (2, 24, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(31−21, 31, 321)-Net in Base 64

(10, 31, 321)-net in base 64, using

1 times m-reduction [i] based on (10, 32, 321)-net in base 64, using
- base change [i] based on digital (2, 24, 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]

(31−21, 31, 18839)-Net in Base 64 — Upper bound on s

There is no (10, 31, 18840)-net in base 64, because

1 times m-reduction [i] would yield (10, 30, 18840)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 1 532940 053226 530931 217715 520233 717272 189458 872291 409778 > 64³⁰ [i]