MinT - Best Known (219−135, 219, s)-Nets in Base 3

Best Known (219−135, 219, s)-Nets in Base 3

(219−135, 219, 59)-Net over F₃ — Constructive and digital

Digital (84, 219, 59)-net over F₃, using

net from sequence [i] based on digital (84, 58)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 58)-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]

(219−135, 219, 84)-Net over F₃ — Digital

Digital (84, 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]

(219−135, 219, 397)-Net in Base 3 — Upper bound on s

There is no (84, 219, 398)-net in base 3, because

1 times m-reduction [i] would yield (84, 218, 398)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 115 318521 787994 237209 967780 213955 846716 466033 217568 533363 947427 538095 258979 533756 148745 970830 660256 588353 > 3²¹⁸ [i]