MinT - Best Known (140−107, 140, s)-Nets in Base 9

Best Known (140−107, 140, s)-Nets in Base 9

(140−107, 140, 81)-Net over F₉ — Constructive and digital

Digital (33, 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]

(140−107, 140, 128)-Net over F₉ — Digital

Digital (33, 140, 128)-net over F₉, using

net from sequence [i] based on digital (33, 127)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 33 and N(F) ≥ 128, using
  - a curve from the manYPoints database [i]

(140−107, 140, 786)-Net in Base 9 — Upper bound on s

There is no (33, 140, 787)-net in base 9, because

1 times m-reduction [i] would yield (33, 139, 787)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 4 405155 303821 241756 616605 164781 760682 154092 529984 713564 534602 013610 199644 066720 881705 725375 618325 584838 151523 745282 972632 331210 653305 > 9¹³⁹ [i]