MinT - Best Known (86−76, 86, s)-Nets in Base 9

Best Known (86−76, 86, s)-Nets in Base 9

(86−76, 86, 40)-Net over F₉ — Constructive and digital

Digital (10, 86, 40)-net over F₉, using

t-expansion [i] based on digital (8, 86, 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]

(86−76, 86, 54)-Net over F₉ — Digital

Digital (10, 86, 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]

(86−76, 86, 99)-Net in Base 9 — Upper bound on s

There is no (10, 86, 100)-net in base 9, because

1 times m-reduction [i] would yield (10, 85, 100)-net in base 9, but
- extracting embedded orthogonal array [i] would yield OA(9⁸⁵, 100, S₉, 75), but
  - the linear programming bound shows that MÂ â‰¥ 693 349003 011434 276309 531891 149773 253542 241992 640047 050289 802748 390016 846562 612470 179184 834411 / 376983 659851 > 9⁸⁵ [i]