MinT - Best Known (40, 40+67, s)-Nets in Base 9

Best Known (40, 40+67, s)-Nets in Base 9

(40, 40+67, 81)-Net over F₉ — Constructive and digital

Digital (40, 107, 81)-net over F₉, using

t-expansion [i] based on digital (32, 107, 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+67, 140)-Net over F₉ — Digital

Digital (40, 107, 140)-net over F₉, using

t-expansion [i] based on digital (39, 107, 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+67, 1891)-Net in Base 9 — Upper bound on s

There is no (40, 107, 1892)-net in base 9, because

1 times m-reduction [i] would yield (40, 106, 1892)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 142185 167633 905274 101671 192039 316494 096866 685581 799177 547429 154177 192207 745383 334296 389298 293951 596321 > 9¹⁰⁶ [i]