MinT - Best Known (174−97, 174, s)-Nets in Base 3

Best Known (174−97, 174, s)-Nets in Base 3

(174−97, 174, 52)-Net over F₃ — Constructive and digital

Digital (77, 174, 52)-net over F₃, using

net from sequence [i] based on digital (77, 51)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 51)-sequence over F₉, using
  - s-reduction based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

(174−97, 174, 84)-Net over F₃ — Digital

Digital (77, 174, 84)-net over F₃, using

t-expansion [i] based on digital (71, 174, 84)-net over F₃, using
- net from sequence [i] based on digital (71, 83)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 71 and N(F) ≥ 84, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(174−97, 174, 445)-Net in Base 3 — Upper bound on s

There is no (77, 174, 446)-net in base 3, because

1 times m-reduction [i] would yield (77, 173, 446)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 36007 288490 782554 057806 597412 878373 569637 657879 877814 852578 932554 258474 818915 068129 > 3¹⁷³ [i]