MinT - Best Known (40, 40+51, s)-Nets in Base 9

Best Known (40, 40+51, s)-Nets in Base 9

(40, 40+51, 84)-Net over F₉ — Constructive and digital

Digital (40, 91, 84)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (2, 27, 20)-net over F₉, using
  - net from sequence [i] based on digital (2, 19)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 2 and N(F) ≥ 20, using
      - a function field by GarcÃa and Quoos [i]
2. digital (13, 64, 64)-net over F₉, using
  - net from sequence [i] based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

(40, 40+51, 88)-Net in Base 9 — Constructive

(40, 91, 88)-net in base 9, using

2 times m-reduction [i] based on (40, 93, 88)-net in base 9, using
- base change [i] based on digital (9, 62, 88)-net over F₂₇, using
  - net from sequence [i] based on digital (9, 87)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 9 and N(F) ≥ 88, using
      - a function field by SÃ©mirat [i]

(40, 40+51, 140)-Net over F₉ — Digital

Digital (40, 91, 140)-net over F₉, using

t-expansion [i] based on digital (39, 91, 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]

(40, 40+51, 3450)-Net in Base 9 — Upper bound on s

There is no (40, 91, 3451)-net in base 9, because

1 times m-reduction [i] would yield (40, 90, 3451)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 76 321693 240606 906382 124692 826073 964890 327393 172399 210883 756613 797132 513355 501834 577433 > 9⁹⁰ [i]