MinT - Best Known (234−133, 234, s)-Nets in Base 3

Best Known (234−133, 234, s)-Nets in Base 3

(234−133, 234, 68)-Net over F₃ — Constructive and digital

Digital (101, 234, 68)-net over F₃, using

net from sequence [i] based on digital (101, 67)-sequence over F₃, using
- base reduction for sequences [i] based on digital (17, 67)-sequence over F₉, using
  - s-reduction based on digital (17, 73)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 17 and N(F) ≥ 74, using
      - a function field by GarcÃa and Quoos [i]

(234−133, 234, 96)-Net over F₃ — Digital

Digital (101, 234, 96)-net over F₃, using

t-expansion [i] based on digital (89, 234, 96)-net over F₃, using
- net from sequence [i] based on digital (89, 95)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 89 and N(F) ≥ 96, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(234−133, 234, 551)-Net in Base 3 — Upper bound on s

There is no (101, 234, 552)-net in base 3, because

1 times m-reduction [i] would yield (101, 233, 552)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1535 740324 326701 916400 480180 927996 872765 083443 031179 923462 867747 526637 534784 181432 966423 111812 641576 226828 574065 > 3²³³ [i]