MinT - Best Known (90−45, 90, s)-Nets in Base 3

Best Known (90−45, 90, s)-Nets in Base 3

(90−45, 90, 48)-Net over F₃ — Constructive and digital

Digital (45, 90, 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]

(90−45, 90, 56)-Net over F₃ — Digital

Digital (45, 90, 56)-net over F₃, using

t-expansion [i] based on digital (40, 90, 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]

(90−45, 90, 364)-Net in Base 3 — Upper bound on s

There is no (45, 90, 365)-net in base 3, because

1 times m-reduction [i] would yield (45, 89, 365)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3 000233 836028 750666 525751 160762 246405 474449 > 3⁸⁹ [i]