MinT - Best Known (222−123, 222, s)-Nets in Base 3

Best Known (222−123, 222, s)-Nets in Base 3

(222−123, 222, 66)-Net over F₃ — Constructive and digital

Digital (99, 222, 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]

(222−123, 222, 96)-Net over F₃ — Digital

Digital (99, 222, 96)-net over F₃, using

t-expansion [i] based on digital (89, 222, 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]

(222−123, 222, 572)-Net in Base 3 — Upper bound on s

There is no (99, 222, 573)-net in base 3, because

1 times m-reduction [i] would yield (99, 221, 573)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2935 441240 447007 731773 837572 006739 040384 816461 483975 199413 365601 375308 682896 633683 172381 885004 873095 668779 > 3²²¹ [i]