MinT - Best Known (6, 40, s)-Nets in Base 64

Best Known (6, 40, s)-Nets in Base 64

(6, 40, 128)-Net over F₆₄ — Constructive and digital

Digital (6, 40, 128)-net over F₆₄, using

t-expansion [i] based on digital (5, 40, 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, 40, 129)-Net in Base 64 — Constructive

(6, 40, 129)-net in base 64, using

2 times m-reduction [i] based on (6, 42, 129)-net in base 64, using
- base change [i] based on digital (0, 36, 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]

(6, 40, 161)-Net over F₆₄ — Digital

Digital (6, 40, 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, 40, 2016)-Net in Base 64 — Upper bound on s

There is no (6, 40, 2017)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 1 769719 813704 801396 475705 770843 048453 486512 559711 246488 388080 654497 667680 > 64⁴⁰ [i]