MinT - Best Known (219−131, 219, s)-Nets in Base 3

Best Known (219−131, 219, s)-Nets in Base 3

(219−131, 219, 63)-Net over F₃ — Constructive and digital

Digital (88, 219, 63)-net over F₃, using

net from sequence [i] based on digital (88, 62)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 62)-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]

(219−131, 219, 84)-Net over F₃ — Digital

Digital (88, 219, 84)-net over F₃, using

t-expansion [i] based on digital (71, 219, 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]

(219−131, 219, 437)-Net in Base 3 — Upper bound on s

There is no (88, 219, 438)-net in base 3, because

1 times m-reduction [i] would yield (88, 218, 438)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 110 063948 419350 235207 345098 826466 634908 802483 424445 902632 777842 675346 029794 132537 272296 386427 920440 599277 > 3²¹⁸ [i]