MinT - Best Known (41, 140, s)-Nets in Base 9

Best Known (41, 140, s)-Nets in Base 9

(41, 140, 81)-Net over F₉ — Constructive and digital

Digital (41, 140, 81)-net over F₉, using

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

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

t-expansion [i] based on digital (39, 140, 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, 140, 1186)-Net in Base 9 — Upper bound on s

There is no (41, 140, 1187)-net in base 9, because

1 times m-reduction [i] would yield (41, 139, 1187)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 4 415223 718130 084300 851006 306085 457655 488279 158851 265325 526052 284320 212909 467229 812799 542814 775606 753334 641329 511121 057663 343128 821145 > 9¹³⁹ [i]