MinT - Best Known (167−91, 167, s)-Nets in Base 3

Best Known (167−91, 167, s)-Nets in Base 3

(167−91, 167, 56)-Net over F₃ — Constructive and digital

Digital (76, 167, 56)-net over F₃, using

1 times m-reduction [i] based on digital (76, 168, 56)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 61, 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 (15, 107, 28)-net over F₃, using
    - net from sequence [i] based on digital (15, 27)-sequence over F₃ (see above)

(167−91, 167, 84)-Net over F₃ — Digital

Digital (76, 167, 84)-net over F₃, using

t-expansion [i] based on digital (71, 167, 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]

(167−91, 167, 464)-Net in Base 3 — Upper bound on s

There is no (76, 167, 465)-net in base 3, because

1 times m-reduction [i] would yield (76, 166, 465)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 16 998590 693798 683898 717213 728377 732352 913743 794804 002623 157407 358852 260093 571571 > 3¹⁶⁶ [i]