MinT - Best Known (189−95, 189, s)-Nets in Base 3

Best Known (189−95, 189, s)-Nets in Base 3

(189−95, 189, 68)-Net over F₃ — Constructive and digital

Digital (94, 189, 68)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (21, 68, 32)-net over F₃, using
  - net from sequence [i] based on digital (21, 31)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 21 and N(F) ≥ 32, using
      - an explicitly constructive algebraic function field [i]
2. digital (26, 121, 36)-net over F₃, using
  - net from sequence [i] based on digital (26, 35)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 26 and N(F) ≥ 36, using
      - an explicitly constructive algebraic function field [i]

(189−95, 189, 96)-Net over F₃ — Digital

Digital (94, 189, 96)-net over F₃, using

t-expansion [i] based on digital (89, 189, 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]

(189−95, 189, 698)-Net in Base 3 — Upper bound on s

There is no (94, 189, 699)-net in base 3, because

1 times m-reduction [i] would yield (94, 188, 699)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 511306 752942 861028 704855 487611 011549 911960 089832 434115 987465 491360 750194 749901 119161 013139 > 3¹⁸⁸ [i]