MinT - Best Known (140−77, 140, s)-Nets in Base 3

Best Known (140−77, 140, s)-Nets in Base 3

(140−77, 140, 48)-Net over F₃ — Constructive and digital

Digital (63, 140, 48)-net over F₃, using

t-expansion [i] based on digital (45, 140, 48)-net over F₃, using
- net from sequence [i] based on digital (45, 47)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 45 and N(F) ≥ 48, using
    - an explicitly constructive algebraic function field [i]

(140−77, 140, 64)-Net over F₃ — Digital

Digital (63, 140, 64)-net over F₃, using

t-expansion [i] based on digital (49, 140, 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]

(140−77, 140, 381)-Net in Base 3 — Upper bound on s

There is no (63, 140, 382)-net in base 3, because

1 times m-reduction [i] would yield (63, 139, 382)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2 146398 466166 815874 097220 929286 830709 611674 453228 247492 352351 443813 > 3¹³⁹ [i]