MinT - Best Known (186−75, 186, s)-Nets in Base 3

Best Known (186−75, 186, s)-Nets in Base 3

(186−75, 186, 148)-Net over F₃ — Constructive and digital

Digital (111, 186, 148)-net over F₃, using

2 times m-reduction [i] based on digital (111, 188, 148)-net over F₃, using
- trace code for nets [i] based on digital (17, 94, 74)-net over F₉, using
  - net from sequence [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]

(186−75, 186, 171)-Net over F₃ — Digital

Digital (111, 186, 171)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(186−75, 186, 1744)-Net in Base 3 — Upper bound on s

There is no (111, 186, 1745)-net in base 3, because

1 times m-reduction [i] would yield (111, 185, 1745)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 18865 256153 978866 998332 380339 078244 042964 007982 197935 746268 422417 815938 345574 360556 297747 > 3¹⁸⁵ [i]