MinT - Best Known (99, 99+135, s)-Nets in Base 3

Best Known (99, 99+135, s)-Nets in Base 3

(99, 99+135, 66)-Net over F₃ — Constructive and digital

Digital (99, 234, 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]

(99, 99+135, 96)-Net over F₃ — Digital

Digital (99, 234, 96)-net over F₃, using

t-expansion [i] based on digital (89, 234, 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]

(99, 99+135, 524)-Net in Base 3 — Upper bound on s

There is no (99, 234, 525)-net in base 3, because

1 times m-reduction [i] would yield (99, 233, 525)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1522 719547 494656 378798 421366 847752 151365 769289 983057 049614 278879 198907 874907 285784 492281 288867 694659 363711 444939 > 3²³³ [i]