MinT - Best Known (81−41, 81, s)-Nets in Base 3

Best Known (81−41, 81, s)-Nets in Base 3

(81−41, 81, 42)-Net over F₃ — Constructive and digital

Digital (40, 81, 42)-net over F₃, using

t-expansion [i] based on digital (39, 81, 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]

(81−41, 81, 56)-Net over F₃ — Digital

Digital (40, 81, 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]

(81−41, 81, 317)-Net in Base 3 — Upper bound on s

There is no (40, 81, 318)-net in base 3, because

1 times m-reduction [i] would yield (40, 80, 318)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 154 226852 801899 158088 612726 732008 298153 > 3⁸⁰ [i]