MinT - Best Known (210−111, 210, s)-Nets in Base 3

Best Known (210−111, 210, s)-Nets in Base 3

(210−111, 210, 66)-Net over F₃ — Constructive and digital

Digital (99, 210, 66)-net over F₃, using

net from sequence [i] based on digital (99, 65)-sequence over F₃, using
- base reduction for sequences [i] based on digital (17, 65)-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]

(210−111, 210, 96)-Net over F₃ — Digital

Digital (99, 210, 96)-net over F₃, using

t-expansion [i] based on digital (89, 210, 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]

(210−111, 210, 640)-Net in Base 3 — Upper bound on s

There is no (99, 210, 641)-net in base 3, because

1 times m-reduction [i] would yield (99, 209, 641)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 5253 824546 854634 730833 631465 946205 408084 412472 610020 207017 448768 541123 026527 343726 101602 463591 637451 > 3²⁰⁹ [i]