MinT - Best Known (40, 40+65, s)-Nets in Base 9

Best Known (40, 40+65, s)-Nets in Base 9

(40, 40+65, 81)-Net over F₉ — Constructive and digital

Digital (40, 105, 81)-net over F₉, using

t-expansion [i] based on digital (32, 105, 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+65, 140)-Net over F₉ — Digital

Digital (40, 105, 140)-net over F₉, using

t-expansion [i] based on digital (39, 105, 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+65, 1999)-Net in Base 9 — Upper bound on s

There is no (40, 105, 2000)-net in base 9, because

1 times m-reduction [i] would yield (40, 104, 2000)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 1762 502707 144143 091064 084117 761372 950498 940257 081530 641570 919358 505632 202876 202654 156709 500192 829441 > 9¹⁰⁴ [i]