MinT - Best Known (231−129, 231, s)-Nets in Base 3

Best Known (231−129, 231, s)-Nets in Base 3

(231−129, 231, 69)-Net over F₃ — Constructive and digital

Digital (102, 231, 69)-net over F₃, using

net from sequence [i] based on digital (102, 68)-sequence over F₃, using
- base reduction for sequences [i] based on digital (17, 68)-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]

(231−129, 231, 104)-Net over F₃ — Digital

Digital (102, 231, 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]

(231−129, 231, 578)-Net in Base 3 — Upper bound on s

There is no (102, 231, 579)-net in base 3, because

1 times m-reduction [i] would yield (102, 230, 579)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 57 339435 542354 573150 251549 336155 887900 000939 575486 598676 743474 557193 963634 711033 899931 461927 129788 779507 321729 > 3²³⁰ [i]