MinT - Best Known (140−81, 140, s)-Nets in Base 9

Best Known (140−81, 140, s)-Nets in Base 9

(140−81, 140, 98)-Net over F₉ — Constructive and digital

Digital (59, 140, 98)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (6, 46, 34)-net over F₉, using
  - net from sequence [i] based on digital (6, 33)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 6 and N(F) ≥ 34, using
      - a function field by SÃ©mirat [i]
2. digital (13, 94, 64)-net over F₉, using
  - net from sequence [i] based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

(140−81, 140, 182)-Net over F₉ — Digital

Digital (59, 140, 182)-net over F₉, using

t-expansion [i] based on digital (50, 140, 182)-net over F₉, using
- net from sequence [i] based on digital (50, 181)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 50 and N(F) ≥ 182, using
    - a curve from the manYPoints database [i]

(140−81, 140, 4055)-Net in Base 9 — Upper bound on s

There is no (59, 140, 4056)-net in base 9, because

1 times m-reduction [i] would yield (59, 139, 4056)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 4 368236 944979 224923 580273 588323 653651 464155 377839 721674 037548 674688 352473 578853 775402 356188 060874 394292 482291 867896 223899 264140 030465 > 9¹³⁹ [i]