MinT - Best Known (67, 67+77, s)-Nets in Base 3

Best Known (67, 67+77, s)-Nets in Base 3

(67, 67+77, 52)-Net over F₃ — Constructive and digital

Digital (67, 144, 52)-net over F₃, using

1 times m-reduction [i] based on digital (67, 145, 52)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (13, 52, 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, 93, 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]

(67, 67+77, 72)-Net over F₃ — Digital

Digital (67, 144, 72)-net over F₃, using

net from sequence [i] based on digital (67, 71)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 67 and N(F) ≥ 72, using
  - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(67, 67+77, 432)-Net in Base 3 — Upper bound on s

There is no (67, 144, 433)-net in base 3, because

1 times m-reduction [i] would yield (67, 143, 433)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 172 108168 584788 539941 238996 673182 293651 474443 067158 679778 291225 244217 > 3¹⁴³ [i]