MinT - Best Known (8−6, 8, s)-Nets in Base 64

Best Known (8−6, 8, s)-Nets in Base 64

(8−6, 8, 80)-Net over F₆₄ — Constructive and digital

Digital (2, 8, 80)-net over F₆₄, using

t-expansion [i] based on digital (1, 8, 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]

(8−6, 8, 97)-Net over F₆₄ — Digital

Digital (2, 8, 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]

(8−6, 8, 257)-Net in Base 64 — Constructive

(2, 8, 257)-net in base 64, using

base change [i] based on digital (0, 6, 257)-net over F₂₅₆, using
- net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
  - generalized Faure sequence [i]
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
    - the rational function field F₂₅₆(x) [i]
  - Niederreiter sequence [i]

(8−6, 8, 1889)-Net in Base 64 — Upper bound on s

There is no (2, 8, 1890)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 281 823792 021136 > 64⁸ [i]