MinT - Best Known (236−133, 236, s)-Nets in Base 3

Best Known (236−133, 236, s)-Nets in Base 3

(236−133, 236, 70)-Net over F₃ — Constructive and digital

Digital (103, 236, 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]

(236−133, 236, 104)-Net over F₃ — Digital

Digital (103, 236, 104)-net over F₃, using

t-expansion [i] based on digital (102, 236, 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]

(236−133, 236, 572)-Net in Base 3 — Upper bound on s

There is no (103, 236, 573)-net in base 3, because

1 times m-reduction [i] would yield (103, 235, 573)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 14223 379433 755622 208576 007433 918359 218672 041070 782755 010646 275780 265710 039493 287297 481849 609540 264040 627558 137729 > 3²³⁵ [i]