MinT - Best Known (230−127, 230, s)-Nets in Base 3

Best Known (230−127, 230, s)-Nets in Base 3

(230−127, 230, 70)-Net over F₃ — Constructive and digital

Digital (103, 230, 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]

(230−127, 230, 104)-Net over F₃ — Digital

Digital (103, 230, 104)-net over F₃, using

t-expansion [i] based on digital (102, 230, 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]

(230−127, 230, 598)-Net in Base 3 — Upper bound on s

There is no (103, 230, 599)-net in base 3, because

1 times m-reduction [i] would yield (103, 229, 599)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 18 355268 356761 039341 457914 968614 821502 463474 333769 162840 953309 263582 271027 390142 770822 598967 236895 788961 365475 > 3²²⁹ [i]