MinT - Best Known (250−139, 250, s)-Nets in Base 3

Best Known (250−139, 250, s)-Nets in Base 3

(250−139, 250, 74)-Net over F₃ — Constructive and digital

Digital (111, 250, 74)-net over F₃, using

t-expansion [i] based on digital (107, 250, 74)-net over F₃, using
- net from sequence [i] based on digital (107, 73)-sequence over F₃, using
  - base reduction for sequences [i] 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]

(250−139, 250, 104)-Net over F₃ — Digital

Digital (111, 250, 104)-net over F₃, using

t-expansion [i] based on digital (102, 250, 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]

(250−139, 250, 633)-Net in Base 3 — Upper bound on s

There is no (111, 250, 634)-net in base 3, because

1 times m-reduction [i] would yield (111, 249, 634)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 69671 569572 859783 482378 312294 178307 254337 114659 536454 870109 463305 826804 101072 268950 347756 572478 895558 792727 402617 643597 > 3²⁴⁹ [i]