MinT - Best Known (166−87, 166, s)-Nets in Base 3

Best Known (166−87, 166, s)-Nets in Base 3

(166−87, 166, 60)-Net over F₃ — Constructive and digital

Digital (79, 166, 60)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (15, 58, 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]
2. digital (21, 108, 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]

(166−87, 166, 84)-Net over F₃ — Digital

Digital (79, 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]

(166−87, 166, 530)-Net in Base 3 — Upper bound on s

There is no (79, 166, 531)-net in base 3, because

1 times m-reduction [i] would yield (79, 165, 531)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 5 480640 105038 577259 534575 657490 427769 077224 375846 004204 278663 992382 905366 938395 > 3¹⁶⁵ [i]