MinT - Best Known (140−97, 140, s)-Nets in Base 9

Best Known (140−97, 140, s)-Nets in Base 9

(140−97, 140, 81)-Net over F₉ — Constructive and digital

Digital (43, 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]

(140−97, 140, 147)-Net over F₉ — Digital

Digital (43, 140, 147)-net over F₉, using

net from sequence [i] based on digital (43, 146)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 43 and N(F) ≥ 147, using
  - a curve from the manYPoints database [i]

(140−97, 140, 1329)-Net in Base 9 — Upper bound on s

There is no (43, 140, 1330)-net in base 9, because

1 times m-reduction [i] would yield (43, 139, 1330)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 4 505733 528470 922239 757851 984391 296798 444099 801627 214267 096134 038783 291871 745480 102653 580543 987676 342558 981482 895307 109630 229273 839873 > 9¹³⁹ [i]