MinT - Best Known (111−55, 111, s)-Nets in Base 3

Best Known (111−55, 111, s)-Nets in Base 3

(111−55, 111, 52)-Net over F₃ — Constructive and digital

Digital (56, 111, 52)-net over F₃, using

1 times m-reduction [i] based on digital (56, 112, 52)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (13, 41, 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, 71, 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]

(111−55, 111, 64)-Net over F₃ — Digital

Digital (56, 111, 64)-net over F₃, using

t-expansion [i] based on digital (49, 111, 64)-net over F₃, using
- net from sequence [i] based on digital (49, 63)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 49 and N(F) ≥ 64, using
    - a curve from the manYPoints database [i]

(111−55, 111, 454)-Net in Base 3 — Upper bound on s

There is no (56, 111, 455)-net in base 3, because

1 times m-reduction [i] would yield (56, 110, 455)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 32098 895889 738626 188220 824002 265405 634391 255542 917739 > 3¹¹⁰ [i]