MinT - Best Known (96, 96+123, s)-Nets in Base 3

Best Known (96, 96+123, s)-Nets in Base 3

(96, 96+123, 64)-Net over F₃ — Constructive and digital

Digital (96, 219, 64)-net over F₃, using

t-expansion [i] based on digital (89, 219, 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]

(96, 96+123, 96)-Net over F₃ — Digital

Digital (96, 219, 96)-net over F₃, using

t-expansion [i] based on digital (89, 219, 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]

(96, 96+123, 539)-Net in Base 3 — Upper bound on s

There is no (96, 219, 540)-net in base 3, because

1 times m-reduction [i] would yield (96, 218, 540)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 109 594120 264072 845070 951097 567466 478923 525560 887126 496484 744268 771039 966729 697689 918185 816890 435440 247577 > 3²¹⁸ [i]