MinT - Best Known (189−106, 189, s)-Nets in Base 3

Best Known (189−106, 189, s)-Nets in Base 3

(189−106, 189, 58)-Net over F₃ — Constructive and digital

Digital (83, 189, 58)-net over F₃, using

net from sequence [i] based on digital (83, 57)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 57)-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]

(189−106, 189, 84)-Net over F₃ — Digital

Digital (83, 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]

(189−106, 189, 467)-Net in Base 3 — Upper bound on s

There is no (83, 189, 468)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 588161 525602 213822 196516 826510 767914 554444 233071 475195 731496 896670 602727 561993 679691 776137 > 3¹⁸⁹ [i]