MinT - Best Known (40, 40+57, s)-Nets in Base 9

Best Known (40, 40+57, s)-Nets in Base 9

(40, 40+57, 81)-Net over F₉ — Constructive and digital

Digital (40, 97, 81)-net over F₉, using

t-expansion [i] based on digital (32, 97, 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]

(40, 40+57, 82)-Net in Base 9 — Constructive

(40, 97, 82)-net in base 9, using

2 times m-reduction [i] based on (40, 99, 82)-net in base 9, using
- base change [i] based on digital (7, 66, 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]

(40, 40+57, 140)-Net over F₉ — Digital

Digital (40, 97, 140)-net over F₉, using

t-expansion [i] based on digital (39, 97, 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+57, 2623)-Net in Base 9 — Upper bound on s

There is no (40, 97, 2624)-net in base 9, because

1 times m-reduction [i] would yield (40, 96, 2624)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 40 902030 534017 268523 507165 572576 008123 826011 240511 305228 919942 432423 442914 223928 862371 629057 > 9⁹⁶ [i]