MinT - Best Known (87−43, 87, s)-Nets in Base 3

Best Known (87−43, 87, s)-Nets in Base 3

(87−43, 87, 44)-Net over F₃ — Constructive and digital

Digital (44, 87, 44)-net over F₃, using

1 times m-reduction [i] based on digital (44, 88, 44)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (7, 29, 16)-net over F₃, using
    - net from sequence [i] based on digital (7, 15)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 7 and N(F) ≥ 16, using
        an explicitly constructive algebraic function field [i]
  2. digital (15, 59, 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]

(87−43, 87, 56)-Net over F₃ — Digital

Digital (44, 87, 56)-net over F₃, using

t-expansion [i] based on digital (40, 87, 56)-net over F₃, using
- net from sequence [i] based on digital (40, 55)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 40 and N(F) ≥ 56, using
    - a curve from the manYPoints database [i]

(87−43, 87, 370)-Net in Base 3 — Upper bound on s

There is no (44, 87, 371)-net in base 3, because

1 times m-reduction [i] would yield (44, 86, 371)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 112321 471314 759312 808782 827722 665232 738623 > 3⁸⁶ [i]