MinT - Best Known (209−108, 209, s)-Nets in Base 3

Best Known (209−108, 209, s)-Nets in Base 3

(209−108, 209, 68)-Net over F₃ — Constructive and digital

Digital (101, 209, 68)-net over F₃, using

net from sequence [i] based on digital (101, 67)-sequence over F₃, using
- base reduction for sequences [i] based on digital (17, 67)-sequence over F₉, using
  - s-reduction based on digital (17, 73)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 17 and N(F) ≥ 74, using
      - a function field by GarcÃa and Quoos [i]

(209−108, 209, 96)-Net over F₃ — Digital

Digital (101, 209, 96)-net over F₃, using

t-expansion [i] based on digital (89, 209, 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]

(209−108, 209, 684)-Net in Base 3 — Upper bound on s

There is no (101, 209, 685)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 5416 579210 755403 038358 215099 398863 599190 927331 294757 771885 240454 241075 760645 548657 931235 445027 393937 > 3²⁰⁹ [i]