MinT - Best Known (238−149, 238, s)-Nets in Base 3

Best Known (238−149, 238, s)-Nets in Base 3

(238−149, 238, 64)-Net over F₃ — Constructive and digital

Digital (89, 238, 64)-net over F₃, using

net from sequence [i] based on digital (89, 63)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 63)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
    - a function field by GarcÃa and Quoos [i]

(238−149, 238, 96)-Net over F₃ — Digital

Digital (89, 238, 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]

(238−149, 238, 409)-Net in Base 3 — Upper bound on s

There is no (89, 238, 410)-net in base 3, because

1 times m-reduction [i] would yield (89, 237, 410)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 129579 540186 677986 003850 514408 102742 009184 007913 486682 802068 496922 451151 957847 056510 860410 085973 643555 904330 971317 > 3²³⁷ [i]