MinT - Best Known (216−109, 216, s)-Nets in Base 3

Best Known (216−109, 216, s)-Nets in Base 3

(216−109, 216, 74)-Net over F₃ — Constructive and digital

Digital (107, 216, 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]

(216−109, 216, 104)-Net over F₃ — Digital

Digital (107, 216, 104)-net over F₃, using

t-expansion [i] based on digital (102, 216, 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]

(216−109, 216, 779)-Net in Base 3 — Upper bound on s

There is no (107, 216, 780)-net in base 3, because

1 times m-reduction [i] would yield (107, 215, 780)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3 821172 155931 751349 214672 002972 007143 321846 739445 355925 913047 803654 515455 032651 432163 599079 570505 738761 > 3²¹⁵ [i]