MinT - Best Known (219−127, 219, s)-Nets in Base 3

Best Known (219−127, 219, s)-Nets in Base 3

(219−127, 219, 64)-Net over F₃ — Constructive and digital

Digital (92, 219, 64)-net over F₃, using

t-expansion [i] based on digital (89, 219, 64)-net over F₃, using
- net from sequence [i] based on digital (89, 63)-sequence over F₃, using
  - base reduction for sequences [i] 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]

(219−127, 219, 96)-Net over F₃ — Digital

Digital (92, 219, 96)-net over F₃, using

t-expansion [i] based on digital (89, 219, 96)-net over F₃, using
- net from sequence [i] based on digital (89, 95)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 89 and N(F) ≥ 96, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(219−127, 219, 484)-Net in Base 3 — Upper bound on s

There is no (92, 219, 485)-net in base 3, because

1 times m-reduction [i] would yield (92, 218, 485)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 110 905778 015392 809526 373142 974347 890822 188121 796425 602638 379042 029705 093412 455252 778085 082688 376442 527931 > 3²¹⁸ [i]