MinT - Best Known (247−139, 247, s)-Nets in Base 3

Best Known (247−139, 247, s)-Nets in Base 3

(247−139, 247, 74)-Net over F₃ — Constructive and digital

Digital (108, 247, 74)-net over F₃, using

t-expansion [i] based on digital (107, 247, 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]

(247−139, 247, 104)-Net over F₃ — Digital

Digital (108, 247, 104)-net over F₃, using

t-expansion [i] based on digital (102, 247, 104)-net over F₃, using
- net from sequence [i] based on digital (102, 103)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 102 and N(F) ≥ 104, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(247−139, 247, 600)-Net in Base 3 — Upper bound on s

There is no (108, 247, 601)-net in base 3, because

1 times m-reduction [i] would yield (108, 246, 601)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2454 967090 175407 154495 408316 582403 139947 668541 548636 231273 041234 801180 523538 716714 412221 580977 597153 598142 987335 462947 > 3²⁴⁶ [i]