MinT - Best Known (227−141, 227, s)-Nets in Base 3

Best Known (227−141, 227, s)-Nets in Base 3

(227−141, 227, 61)-Net over F₃ — Constructive and digital

Digital (86, 227, 61)-net over F₃, using

net from sequence [i] based on digital (86, 60)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 60)-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]

(227−141, 227, 84)-Net over F₃ — Digital

Digital (86, 227, 84)-net over F₃, using

t-expansion [i] based on digital (71, 227, 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]

(227−141, 227, 401)-Net in Base 3 — Upper bound on s

There is no (86, 227, 402)-net in base 3, because

1 times m-reduction [i] would yield (86, 226, 402)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 757756 336099 407652 053467 749307 448143 145874 246194 550799 591784 908200 870995 353800 348928 889795 774824 513326 373661 > 3²²⁶ [i]