MinT - Best Known (231−139, 231, s)-Nets in Base 3

Best Known (231−139, 231, s)-Nets in Base 3

(231−139, 231, 64)-Net over F₃ — Constructive and digital

Digital (92, 231, 64)-net over F₃, using

t-expansion [i] based on digital (89, 231, 64)-net over F₃, using
- net from sequence [i] based on digital (89, 63)-sequence over F₃, using
  - base reduction for sequences [i] based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

(231−139, 231, 96)-Net over F₃ — Digital

Digital (92, 231, 96)-net over F₃, using

t-expansion [i] based on digital (89, 231, 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]

(231−139, 231, 451)-Net in Base 3 — Upper bound on s

There is no (92, 231, 452)-net in base 3, because

1 times m-reduction [i] would yield (92, 230, 452)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 58 486599 307953 444159 238471 917436 700012 757650 061782 527867 702445 585081 912057 523873 532115 727799 007752 637271 959913 > 3²³⁰ [i]