MinT - Best Known (183−91, 183, s)-Nets in Base 3

Best Known (183−91, 183, s)-Nets in Base 3

(183−91, 183, 68)-Net over F₃ — Constructive and digital

Digital (92, 183, 68)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (21, 66, 32)-net over F₃, using
  - net from sequence [i] based on digital (21, 31)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 21 and N(F) ≥ 32, using
      - an explicitly constructive algebraic function field [i]
2. digital (26, 117, 36)-net over F₃, using
  - net from sequence [i] based on digital (26, 35)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 26 and N(F) ≥ 36, using
      - an explicitly constructive algebraic function field [i]

(183−91, 183, 96)-Net over F₃ — Digital

Digital (92, 183, 96)-net over F₃, using

t-expansion [i] based on digital (89, 183, 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]

(183−91, 183, 706)-Net in Base 3 — Upper bound on s

There is no (92, 183, 707)-net in base 3, because

1 times m-reduction [i] would yield (92, 182, 707)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 720 440860 228592 973432 848718 353856 475975 673103 741584 960599 633703 208845 789117 732179 032303 > 3¹⁸² [i]