MinT - Best Known (212−109, 212, s)-Nets in Base 3

Best Known (212−109, 212, s)-Nets in Base 3

(212−109, 212, 70)-Net over F₃ — Constructive and digital

Digital (103, 212, 70)-net over F₃, using

net from sequence [i] based on digital (103, 69)-sequence over F₃, using
- base reduction for sequences [i] based on digital (17, 69)-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]

(212−109, 212, 104)-Net over F₃ — Digital

Digital (103, 212, 104)-net over F₃, using

t-expansion [i] based on digital (102, 212, 104)-net over F₃, using
- net from sequence [i] based on digital (102, 103)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 102 and N(F) ≥ 104, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(212−109, 212, 715)-Net in Base 3 — Upper bound on s

There is no (103, 212, 716)-net in base 3, because

1 times m-reduction [i] would yield (103, 211, 716)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 50339 903501 756569 179859 599099 742237 639711 484015 521687 954117 484264 617270 162852 250534 114289 295260 569737 > 3²¹¹ [i]