MinT - Best Known (86−51, 86, s)-Nets in Base 3

Best Known (86−51, 86, s)-Nets in Base 3

(86−51, 86, 38)-Net over F₃ — Constructive and digital

Digital (35, 86, 38)-net over F₃, using

t-expansion [i] based on digital (32, 86, 38)-net over F₃, using
- net from sequence [i] based on digital (32, 37)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 32 and N(F) ≥ 38, using
    - an explicitly constructive algebraic function field [i]

(86−51, 86, 47)-Net over F₃ — Digital

Digital (35, 86, 47)-net over F₃, using

net from sequence [i] based on digital (35, 46)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 35 and N(F) ≥ 47, using
  - a curve from the manYPoints database [i]

(86−51, 86, 189)-Net in Base 3 — Upper bound on s

There is no (35, 86, 190)-net in base 3, because

1 times m-reduction [i] would yield (35, 85, 190)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 36691 542281 996732 055321 216918 584235 713997 > 3⁸⁵ [i]