MinT - Best Known (9−7, 9, s)-Nets in Base 64

Best Known (9−7, 9, s)-Nets in Base 64

(9−7, 9, 80)-Net over F₆₄ — Constructive and digital

Digital (2, 9, 80)-net over F₆₄, using

t-expansion [i] based on digital (1, 9, 80)-net over F₆₄, using
- net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
    - a function field by SÃ©mirat [i]

(9−7, 9, 97)-Net over F₆₄ — Digital

Digital (2, 9, 97)-net over F₆₄, using

net from sequence [i] based on digital (2, 96)-sequence over F₆₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 2 and N(F) ≥ 97, using
  - sharp bound on the number of rational points on curves with genus 2 [i]

(9−7, 9, 129)-Net in Base 64 — Constructive

(2, 9, 129)-net in base 64, using

5 times m-reduction [i] based on (2, 14, 129)-net in base 64, using
- base change [i] based on digital (0, 12, 129)-net over F₁₂₈, using
  - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
      - the rational function field F₁₂₈(x) [i]
    - Niederreiter sequence [i]

(9−7, 9, 1889)-Net in Base 64 — Upper bound on s

There is no (2, 9, 1890)-net in base 64, because

1 times m-reduction [i] would yield (2, 8, 1890)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 281 823792 021136 > 64⁸ [i]