MinT - Best Known (230−139, 230, s)-Nets in Base 3

Best Known (230−139, 230, s)-Nets in Base 3

(230−139, 230, 64)-Net over F₃ — Constructive and digital

Digital (91, 230, 64)-net over F₃, using

t-expansion [i] based on digital (89, 230, 64)-net over F₃, using
- net from sequence [i] based on digital (89, 63)-sequence over F₃, using
  - base reduction for sequences [i] 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]

(230−139, 230, 96)-Net over F₃ — Digital

Digital (91, 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]

(230−139, 230, 443)-Net in Base 3 — Upper bound on s

There is no (91, 230, 444)-net in base 3, because

1 times m-reduction [i] would yield (91, 229, 444)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 19 799049 054654 943518 516078 605755 779380 842283 961595 368921 742040 481442 105780 110888 220457 186249 621778 038158 656345 > 3²²⁹ [i]