MinT - Best Known (226−113, 226, s)-Nets in Base 3

Best Known (226−113, 226, s)-Nets in Base 3

(226−113, 226, 74)-Net over F₃ — Constructive and digital

Digital (113, 226, 74)-net over F₃, using

t-expansion [i] based on digital (107, 226, 74)-net over F₃, using
- net from sequence [i] based on digital (107, 73)-sequence over F₃, using
  - base reduction for sequences [i] 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]

(226−113, 226, 120)-Net over F₃ — Digital

Digital (113, 226, 120)-net over F₃, using

net from sequence [i] based on digital (113, 119)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 113 and N(F) ≥ 120, using
  - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(226−113, 226, 842)-Net in Base 3 — Upper bound on s

There is no (113, 226, 843)-net in base 3, because

1 times m-reduction [i] would yield (113, 225, 843)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 231911 687665 513285 881329 285810 547906 262024 915960 692306 677425 947067 372654 897199 772778 775782 881553 380328 205841 > 3²²⁵ [i]