MinT - Best Known (237−137, 237, s)-Nets in Base 3

Best Known (237−137, 237, s)-Nets in Base 3

(237−137, 237, 67)-Net over F₃ — Constructive and digital

Digital (100, 237, 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]

(237−137, 237, 96)-Net over F₃ — Digital

Digital (100, 237, 96)-net over F₃, using

t-expansion [i] based on digital (89, 237, 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]

(237−137, 237, 527)-Net in Base 3 — Upper bound on s

There is no (100, 237, 528)-net in base 3, because

1 times m-reduction [i] would yield (100, 236, 528)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 41455 841459 111360 277922 044054 904269 297837 499689 698457 855789 942789 353586 220331 636701 076059 964154 440688 491825 780097 > 3²³⁶ [i]