MinT - Best Known (216−127, 216, s)-Nets in Base 3

Best Known (216−127, 216, s)-Nets in Base 3

(216−127, 216, 64)-Net over F₃ — Constructive and digital

Digital (89, 216, 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]

(216−127, 216, 96)-Net over F₃ — Digital

Digital (89, 216, 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]

(216−127, 216, 456)-Net in Base 3 — Upper bound on s

There is no (89, 216, 457)-net in base 3, because

1 times m-reduction [i] would yield (89, 215, 457)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3 921089 893003 630912 636163 052460 555425 132967 661184 465253 947432 702845 432126 382070 748383 457721 082273 222347 > 3²¹⁵ [i]