MinT - Best Known (142−81, 142, s)-Nets in Base 9

Best Known (142−81, 142, s)-Nets in Base 9

(142−81, 142, 104)-Net over F₉ — Constructive and digital

Digital (61, 142, 104)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (8, 48, 40)-net over F₉, using
  - net from sequence [i] based on digital (8, 39)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 8 and N(F) ≥ 40, 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]

(142−81, 142, 192)-Net over F₉ — Digital

Digital (61, 142, 192)-net over F₉, using

net from sequence [i] based on digital (61, 191)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 61 and N(F) ≥ 192, using
  - Drinfeld modules of rank 1 [i]

(142−81, 142, 4529)-Net in Base 9 — Upper bound on s

There is no (61, 142, 4530)-net in base 9, because

1 times m-reduction [i] would yield (61, 141, 4530)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 354 352360 108075 063149 168058 573468 805317 650805 275241 490991 267266 619276 132557 479282 330337 825660 667059 115266 603719 291776 336418 478836 756353 > 9¹⁴¹ [i]