MinT - Best Known (166−95, 166, s)-Nets in Base 3

Best Known (166−95, 166, s)-Nets in Base 3

(166−95, 166, 48)-Net over F₃ — Constructive and digital

Digital (71, 166, 48)-net over F₃, using

t-expansion [i] based on digital (45, 166, 48)-net over F₃, using
- net from sequence [i] based on digital (45, 47)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 45 and N(F) ≥ 48, using
    - an explicitly constructive algebraic function field [i]

(166−95, 166, 84)-Net over F₃ — Digital

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]

(166−95, 166, 390)-Net in Base 3 — Upper bound on s

There is no (71, 166, 391)-net in base 3, because

1 times m-reduction [i] would yield (71, 165, 391)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 5 891459 343207 570344 337019 821797 078160 969038 979615 448515 160772 310328 952351 997891 > 3¹⁶⁵ [i]