MinT - Best Known (140−87, 140, s)-Nets in Base 9

Best Known (140−87, 140, s)-Nets in Base 9

(140−87, 140, 81)-Net over F₉ — Constructive and digital

Digital (53, 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]

(140−87, 140, 182)-Net over F₉ — Digital

Digital (53, 140, 182)-net over F₉, using

t-expansion [i] based on digital (50, 140, 182)-net over F₉, using
- net from sequence [i] based on digital (50, 181)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 50 and N(F) ≥ 182, using
    - a curve from the manYPoints database [i]

(140−87, 140, 2538)-Net in Base 9 — Upper bound on s

There is no (53, 140, 2539)-net in base 9, because

1 times m-reduction [i] would yield (53, 139, 2539)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 4 411976 552094 635365 860753 600157 052976 579427 364853 754097 848067 037906 354742 291265 024617 163088 330301 285246 331510 435637 478022 793979 834185 > 9¹³⁹ [i]