MinT - Best Known (231−131, 231, s)-Nets in Base 3

Best Known (231−131, 231, s)-Nets in Base 3

(231−131, 231, 67)-Net over F₃ — Constructive and digital

Digital (100, 231, 67)-net over F₃, using

net from sequence [i] based on digital (100, 66)-sequence over F₃, using
- base reduction for sequences [i] based on digital (17, 66)-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]

(231−131, 231, 96)-Net over F₃ — Digital

Digital (100, 231, 96)-net over F₃, using

t-expansion [i] based on digital (89, 231, 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]

(231−131, 231, 549)-Net in Base 3 — Upper bound on s

There is no (100, 231, 550)-net in base 3, because

1 times m-reduction [i] would yield (100, 230, 550)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 60 424405 995944 258526 404464 798079 425895 586456 265502 602011 349117 200473 009378 153202 082214 192255 566075 094492 132301 > 3²³⁰ [i]