MinT - Best Known (41, 41+69, s)-Nets in Base 9

Best Known (41, 41+69, s)-Nets in Base 9

(41, 41+69, 81)-Net over F₉ — Constructive and digital

Digital (41, 110, 81)-net over F₉, using

t-expansion [i] based on digital (32, 110, 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]

(41, 41+69, 140)-Net over F₉ — Digital

Digital (41, 110, 140)-net over F₉, using

t-expansion [i] based on digital (39, 110, 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]

(41, 41+69, 1918)-Net in Base 9 — Upper bound on s

There is no (41, 110, 1919)-net in base 9, because

1 times m-reduction [i] would yield (41, 109, 1919)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 104 172823 889849 001390 413958 736416 372177 371661 512491 882690 301181 265473 002941 588140 808541 742949 412131 440369 > 9¹⁰⁹ [i]