MinT - Best Known (6, 6+21, s)-Nets in Base 64

Best Known (6, 6+21, s)-Nets in Base 64

(6, 6+21, 128)-Net over F₆₄ — Constructive and digital

Digital (6, 27, 128)-net over F₆₄, using

t-expansion [i] based on digital (5, 27, 128)-net over F₆₄, using
- net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
    - a function field by SÃ©mirat [i]

(6, 6+21, 150)-Net in Base 64 — Constructive

(6, 27, 150)-net in base 64, using

8 times m-reduction [i] based on (6, 35, 150)-net in base 64, using
- base change [i] based on digital (1, 30, 150)-net over F₁₂₈, using
  - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
      - a function field by SÃ©mirat [i]

(6, 6+21, 161)-Net over F₆₄ — Digital

Digital (6, 27, 161)-net over F₆₄, using

net from sequence [i] based on digital (6, 160)-sequence over F₆₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 6 and N(F) ≥ 161, using
  - a curve from the manYPoints database [i]

(6, 6+21, 172)-Net in Base 64

(6, 27, 172)-net in base 64, using

1 times m-reduction [i] based on (6, 28, 172)-net in base 64, using
- base change [i] based on digital (2, 24, 172)-net over F₁₂₈, using
  - net from sequence [i] based on digital (2, 171)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 2 and N(F) ≥ 172, using
      - sharp bound on the number of rational points on curves with genus 2 [i]

(6, 6+21, 3565)-Net in Base 64 — Upper bound on s

There is no (6, 27, 3566)-net in base 64, because

1 times m-reduction [i] would yield (6, 26, 3566)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 91437 480183 167207 649887 537645 999877 685429 832230 > 64²⁶ [i]