MinT - Best Known (219−138, 219, s)-Nets in Base 3

Best Known (219−138, 219, s)-Nets in Base 3

(219−138, 219, 56)-Net over F₃ — Constructive and digital

Digital (81, 219, 56)-net over F₃, using

net from sequence [i] based on digital (81, 55)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 55)-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]

(219−138, 219, 84)-Net over F₃ — Digital

Digital (81, 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]

(219−138, 219, 369)-Net in Base 3 — Upper bound on s

There is no (81, 219, 370)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 353 965184 846535 116160 767381 146914 808138 888497 593259 330960 094116 774302 857227 655293 367876 456650 830088 622845 > 3²¹⁹ [i]