MinT - Best Known (96−57, 96, s)-Nets in Base 9

Best Known (96−57, 96, s)-Nets in Base 9

(96−57, 96, 81)-Net over F₉ — Constructive and digital

Digital (39, 96, 81)-net over F₉, using

t-expansion [i] based on digital (32, 96, 81)-net over F₉, using
- net from sequence [i] based on digital (32, 80)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 32 and N(F) ≥ 81, using
    - F₄ from the tower of function fields by Bezerra and GarcÃa over F₉ [i]

(96−57, 96, 82)-Net in Base 9 — Constructive

(39, 96, 82)-net in base 9, using

base change [i] based on digital (7, 64, 82)-net over F₂₇, using
- net from sequence [i] based on digital (7, 81)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 7 and N(F) ≥ 82, using
    - a function field by SÃ©mirat [i]

(96−57, 96, 140)-Net over F₉ — Digital

Digital (39, 96, 140)-net over F₉, using

net from sequence [i] based on digital (39, 139)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 39 and N(F) ≥ 140, using
  - a curve from the manYPoints database [i]

(96−57, 96, 2423)-Net in Base 9 — Upper bound on s

There is no (39, 96, 2424)-net in base 9, because

1 times m-reduction [i] would yield (39, 95, 2424)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 4 509226 462606 375189 293923 916924 241109 976979 997907 413896 690174 192293 868714 843856 743698 816769 > 9⁹⁵ [i]