MinT - Best Known (218−117, 218, s)-Nets in Base 3

Best Known (218−117, 218, s)-Nets in Base 3

(218−117, 218, 68)-Net over F₃ — Constructive and digital

Digital (101, 218, 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]

(218−117, 218, 96)-Net over F₃ — Digital

Digital (101, 218, 96)-net over F₃, using

t-expansion [i] based on digital (89, 218, 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]

(218−117, 218, 628)-Net in Base 3 — Upper bound on s

There is no (101, 218, 629)-net in base 3, because

1 times m-reduction [i] would yield (101, 217, 629)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 34 914745 996732 660754 620822 516130 996088 964952 929740 474087 900382 370719 803766 724578 728115 616474 547647 609105 > 3²¹⁷ [i]