MinT - Best Known (86−33, 86, s)-Nets in Base 3

Best Known (86−33, 86, s)-Nets in Base 3

(86−33, 86, 80)-Net over F₃ — Constructive and digital

Digital (53, 86, 80)-net over F₃, using

4 times m-reduction [i] based on digital (53, 90, 80)-net over F₃, using
- trace code for nets [i] based on digital (8, 45, 40)-net over F₉, using
  - net from sequence [i] based on digital (8, 39)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 8 and N(F) ≥ 40, using
      - a function field by SÃ©mirat [i]

(86−33, 86, 108)-Net over F₃ — Digital

Digital (53, 86, 108)-net over F₃, using

trace code for nets [i] based on digital (10, 43, 54)-net over F₉, using
- net from sequence [i] based on digital (10, 53)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 10 and N(F) ≥ 54, using
    - a curve from the manYPoints database [i]

(86−33, 86, 1149)-Net in Base 3 — Upper bound on s

There is no (53, 86, 1150)-net in base 3, because

1 times m-reduction [i] would yield (53, 85, 1150)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 36269 839336 932604 309539 264968 969910 350241 > 3⁸⁵ [i]