MinT - Best Known (141−101, 141, s)-Nets in Base 9

Best Known (141−101, 141, s)-Nets in Base 9

(141−101, 141, 81)-Net over F₉ — Constructive and digital

Digital (40, 141, 81)-net over F₉, using

t-expansion [i] based on digital (32, 141, 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]

(141−101, 141, 140)-Net over F₉ — Digital

Digital (40, 141, 140)-net over F₉, using

t-expansion [i] based on digital (39, 141, 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]

(141−101, 141, 1113)-Net in Base 9 — Upper bound on s

There is no (40, 141, 1114)-net in base 9, because

1 times m-reduction [i] would yield (40, 140, 1114)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 40 033584 956322 249680 416516 482923 980169 252948 376302 767831 410266 291519 002403 746909 068988 604979 579404 109101 408361 779199 666385 292436 560481 > 9¹⁴⁰ [i]