MinT - Best Known (230−129, 230, s)-Nets in Base 3

Best Known (230−129, 230, s)-Nets in Base 3

(230−129, 230, 68)-Net over F₃ — Constructive and digital

Digital (101, 230, 68)-net over F₃, using

net from sequence [i] based on digital (101, 67)-sequence over F₃, using
- base reduction for sequences [i] based on digital (17, 67)-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]

(230−129, 230, 96)-Net over F₃ — Digital

Digital (101, 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]

(230−129, 230, 567)-Net in Base 3 — Upper bound on s

There is no (101, 230, 568)-net in base 3, because

1 times m-reduction [i] would yield (101, 229, 568)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 18 825751 788975 471609 310134 485328 296060 785524 490085 202664 344916 844076 477693 212072 097933 550480 188166 317846 425601 > 3²²⁹ [i]