MinT - Best Known (189−109, 189, s)-Nets in Base 3

Best Known (189−109, 189, s)-Nets in Base 3

(189−109, 189, 55)-Net over F₃ — Constructive and digital

Digital (80, 189, 55)-net over F₃, using

net from sequence [i] based on digital (80, 54)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 54)-sequence over F₉, using
  - s-reduction 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]

(189−109, 189, 84)-Net over F₃ — Digital

Digital (80, 189, 84)-net over F₃, using

t-expansion [i] based on digital (71, 189, 84)-net over F₃, using
- net from sequence [i] based on digital (71, 83)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 71 and N(F) ≥ 84, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(189−109, 189, 429)-Net in Base 3 — Upper bound on s

There is no (80, 189, 430)-net in base 3, because

1 times m-reduction [i] would yield (80, 188, 430)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 544544 795495 923059 395019 157863 394626 691977 756796 088577 483764 541890 808357 134430 813028 903077 > 3¹⁸⁸ [i]