MinT - Best Known (10−8, 10, s)-Nets in Base 64

Best Known (10−8, 10, s)-Nets in Base 64

(10−8, 10, 80)-Net over F₆₄ — Constructive and digital

Digital (2, 10, 80)-net over F₆₄, using

t-expansion [i] based on digital (1, 10, 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]

(10−8, 10, 97)-Net over F₆₄ — Digital

Digital (2, 10, 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]

(10−8, 10, 129)-Net in Base 64 — Constructive

(2, 10, 129)-net in base 64, using

4 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]

(10−8, 10, 1149)-Net in Base 64 — Upper bound on s

There is no (2, 10, 1150)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 1 154254 463808 922351 > 64¹⁰ [i]