MinT - Best Known (239−139, 239, s)-Nets in Base 3

Best Known (239−139, 239, s)-Nets in Base 3

(239−139, 239, 67)-Net over F₃ — Constructive and digital

Digital (100, 239, 67)-net over F₃, using

net from sequence [i] based on digital (100, 66)-sequence over F₃, using
- base reduction for sequences [i] based on digital (17, 66)-sequence over F₉, using
  - s-reduction based on digital (17, 73)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 17 and N(F) ≥ 74, using
      - a function field by GarcÃa and Quoos [i]

(239−139, 239, 96)-Net over F₃ — Digital

Digital (100, 239, 96)-net over F₃, using

t-expansion [i] based on digital (89, 239, 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]

(239−139, 239, 521)-Net in Base 3 — Upper bound on s

There is no (100, 239, 522)-net in base 3, because

1 times m-reduction [i] would yield (100, 238, 522)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 391502 487704 370073 456176 680909 594247 956664 625579 033709 713499 749444 971855 818262 402026 100075 849976 463190 192591 277293 > 3²³⁸ [i]