MinT - Best Known (129−87, 129, s)-Nets in Base 9

Best Known (129−87, 129, s)-Nets in Base 9

(129−87, 129, 81)-Net over F₉ — Constructive and digital

Digital (42, 129, 81)-net over F₉, using

t-expansion [i] based on digital (32, 129, 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]

(129−87, 129, 140)-Net over F₉ — Digital

Digital (42, 129, 140)-net over F₉, using

t-expansion [i] based on digital (39, 129, 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]

(129−87, 129, 1435)-Net in Base 9 — Upper bound on s

There is no (42, 129, 1436)-net in base 9, because

1 times m-reduction [i] would yield (42, 128, 1436)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 140 436633 813677 633761 922658 563083 889120 790018 370440 973546 773272 567528 583328 223042 742094 189117 873915 858970 789238 821745 334305 > 9¹²⁸ [i]