MinT - Best Known (212−111, 212, s)-Nets in Base 3

Best Known (212−111, 212, s)-Nets in Base 3

(212−111, 212, 68)-Net over F₃ — Constructive and digital

Digital (101, 212, 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]

(212−111, 212, 96)-Net over F₃ — Digital

Digital (101, 212, 96)-net over F₃, using

t-expansion [i] based on digital (89, 212, 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]

(212−111, 212, 669)-Net in Base 3 — Upper bound on s

There is no (101, 212, 670)-net in base 3, because

1 times m-reduction [i] would yield (101, 211, 670)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 50209 490237 434585 386620 404179 003663 662258 725745 588278 789065 656459 699308 616111 680415 590159 561408 570329 > 3²¹¹ [i]