MinT - Best Known (149−76, 149, s)-Nets in Base 3

Best Known (149−76, 149, s)-Nets in Base 3

(149−76, 149, 56)-Net over F₃ — Constructive and digital

Digital (73, 149, 56)-net over F₃, using

10 times m-reduction [i] based on digital (73, 159, 56)-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 (15, 101, 28)-net over F₃, using
    - net from sequence [i] based on digital (15, 27)-sequence over F₃ (see above)

(149−76, 149, 84)-Net over F₃ — Digital

Digital (73, 149, 84)-net over F₃, using

t-expansion [i] based on digital (71, 149, 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]

(149−76, 149, 521)-Net in Base 3 — Upper bound on s

There is no (73, 149, 522)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 127966 737627 844866 865388 672919 815080 209143 731365 415153 174259 949276 380973 > 3¹⁴⁹ [i]