MinT - Best Known (219−133, 219, s)-Nets in Base 3

Best Known (219−133, 219, s)-Nets in Base 3

(219−133, 219, 61)-Net over F₃ — Constructive and digital

Digital (86, 219, 61)-net over F₃, using

net from sequence [i] based on digital (86, 60)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 60)-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]

(219−133, 219, 84)-Net over F₃ — Digital

Digital (86, 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]

(219−133, 219, 416)-Net in Base 3 — Upper bound on s

There is no (86, 219, 417)-net in base 3, because

1 times m-reduction [i] would yield (86, 218, 417)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 108 604335 162354 339866 246821 156794 622310 138888 146357 271006 662231 731016 942694 783974 938469 894296 176270 396745 > 3²¹⁸ [i]