MinT - Best Known (209−111, 209, s)-Nets in Base 3

Best Known (209−111, 209, s)-Nets in Base 3

(209−111, 209, 65)-Net over F₃ — Constructive and digital

Digital (98, 209, 65)-net over F₃, using

net from sequence [i] based on digital (98, 64)-sequence over F₃, using
- base reduction for sequences [i] based on digital (17, 64)-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]

(209−111, 209, 96)-Net over F₃ — Digital

Digital (98, 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]

(209−111, 209, 627)-Net in Base 3 — Upper bound on s

There is no (98, 209, 628)-net in base 3, because

1 times m-reduction [i] would yield (98, 208, 628)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1851 638332 296722 369416 831701 646352 134270 404882 112933 565703 605289 928338 682155 794172 883521 982933 851057 > 3²⁰⁸ [i]