MinT - Best Known (140−91, 140, s)-Nets in Base 9

Best Known (140−91, 140, s)-Nets in Base 9

(140−91, 140, 81)-Net over F₉ — Constructive and digital

Digital (49, 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]

(140−91, 140, 168)-Net over F₉ — Digital

Digital (49, 140, 168)-net over F₉, using

net from sequence [i] based on digital (49, 167)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 49 and N(F) ≥ 168, using
  - Drinfeld modules of rank 1 [i]

(140−91, 140, 1925)-Net in Base 9 — Upper bound on s

There is no (49, 140, 1926)-net in base 9, because

1 times m-reduction [i] would yield (49, 139, 1926)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 4 445835 563344 200331 515757 618929 591588 869714 153147 102115 159058 932637 957048 374761 004369 602655 742903 901614 027636 638413 875183 307425 989105 > 9¹³⁹ [i]