MinT - Best Known (87−45, 87, s)-Nets in Base 3

Best Known (87−45, 87, s)-Nets in Base 3

(87−45, 87, 42)-Net over F₃ — Constructive and digital

Digital (42, 87, 42)-net over F₃, using

t-expansion [i] based on digital (39, 87, 42)-net over F₃, using
- net from sequence [i] based on digital (39, 41)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 39 and N(F) ≥ 42, using
    - an explicitly constructive algebraic function field [i]

(87−45, 87, 56)-Net over F₃ — Digital

Digital (42, 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]

(87−45, 87, 310)-Net in Base 3 — Upper bound on s

There is no (42, 87, 311)-net in base 3, because

1 times m-reduction [i] would yield (42, 86, 311)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 108423 586048 696129 483454 505167 544818 790021 > 3⁸⁶ [i]