MinT - Best Known (140−111, 140, s)-Nets in Base 9

Best Known (140−111, 140, s)-Nets in Base 9

(140−111, 140, 78)-Net over F₉ — Constructive and digital

Digital (29, 140, 78)-net over F₉, using

t-expansion [i] based on digital (22, 140, 78)-net over F₉, using
- net from sequence [i] based on digital (22, 77)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 22 and N(F) ≥ 78, using
    - F₃ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(140−111, 140, 110)-Net over F₉ — Digital

Digital (29, 140, 110)-net over F₉, using

t-expansion [i] based on digital (26, 140, 110)-net over F₉, using
- net from sequence [i] based on digital (26, 109)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 26 and N(F) ≥ 110, using
    - a curve from the manYPoints database [i]

(140−111, 140, 654)-Net in Base 9 — Upper bound on s

There is no (29, 140, 655)-net in base 9, because

1 times m-reduction [i] would yield (29, 139, 655)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 4 440627 350076 279870 831292 715035 764008 859336 919550 868629 710334 625902 128810 904174 414450 524263 031866 198141 545609 730780 405574 740161 858505 > 9¹³⁹ [i]