MinT - Best Known (70−23, 70, s)-Nets in Base 3

Best Known (70−23, 70, s)-Nets in Base 3

(70−23, 70, 84)-Net over F₃ — Constructive and digital

Digital (47, 70, 84)-net over F₃, using

3¹ times duplication [i] based on digital (46, 69, 84)-net over F₃, using
- trace code for nets [i] based on digital (0, 23, 28)-net over F₂₇, using
  - net from sequence [i] based on digital (0, 27)-sequence over F₂₇, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 0 and N(F) ≥ 28, using
      - the rational function field F₂₇(x) [i]
    - Niederreiter sequence [i]

(70−23, 70, 145)-Net over F₃ — Digital

Digital (47, 70, 145)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(70−23, 70, 2404)-Net in Base 3 — Upper bound on s

There is no (47, 70, 2405)-net in base 3, because

1 times m-reduction [i] would yield (47, 69, 2405)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 837 941903 977498 200196 525213 675659 > 3⁶⁹ [i]