MinT - Best Known (230−131, 230, s)-Nets in Base 3

Best Known (230−131, 230, s)-Nets in Base 3

(230−131, 230, 66)-Net over F₃ — Constructive and digital

Digital (99, 230, 66)-net over F₃, using

net from sequence [i] based on digital (99, 65)-sequence over F₃, using
- base reduction for sequences [i] based on digital (17, 65)-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]

(230−131, 230, 96)-Net over F₃ — Digital

Digital (99, 230, 96)-net over F₃, using

t-expansion [i] based on digital (89, 230, 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]

(230−131, 230, 538)-Net in Base 3 — Upper bound on s

There is no (99, 230, 539)-net in base 3, because

1 times m-reduction [i] would yield (99, 229, 539)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 18 487227 431129 703528 592877 105185 659881 453075 555574 549913 665354 173302 099081 971095 237322 031168 099168 142552 671671 > 3²²⁹ [i]