MinT - Best Known (166−92, 166, s)-Nets in Base 3

Best Known (166−92, 166, s)-Nets in Base 3

(166−92, 166, 52)-Net over F₃ — Constructive and digital

Digital (74, 166, 52)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (13, 59, 24)-net over F₃, using
  - net from sequence [i] based on digital (13, 23)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 13 and N(F) ≥ 24, using
      - a function field by SÃ©mirat [i]
2. digital (15, 107, 28)-net over F₃, using
  - net from sequence [i] based on digital (15, 27)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 15 and N(F) ≥ 28, using
      - an explicitly constructive algebraic function field [i]

(166−92, 166, 84)-Net over F₃ — Digital

Digital (74, 166, 84)-net over F₃, using

t-expansion [i] based on digital (71, 166, 84)-net over F₃, using
- net from sequence [i] based on digital (71, 83)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 71 and N(F) ≥ 84, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(166−92, 166, 430)-Net in Base 3 — Upper bound on s

There is no (74, 166, 431)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 16 792514 196002 234930 642179 264109 451794 090843 297778 610814 713137 054487 648748 075397 > 3¹⁶⁶ [i]