MinT - Best Known (236−135, 236, s)-Nets in Base 3

Best Known (236−135, 236, s)-Nets in Base 3

(236−135, 236, 68)-Net over F₃ — Constructive and digital

Digital (101, 236, 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]

(236−135, 236, 96)-Net over F₃ — Digital

Digital (101, 236, 96)-net over F₃, using

t-expansion [i] based on digital (89, 236, 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]

(236−135, 236, 544)-Net in Base 3 — Upper bound on s

There is no (101, 236, 545)-net in base 3, because

1 times m-reduction [i] would yield (101, 235, 545)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 14433 698231 387910 796387 732708 782204 545080 894291 201118 210188 056325 949028 721990 880981 037237 323886 059039 459312 258715 > 3²³⁵ [i]