MinT - Best Known (109−59, 109, s)-Nets in Base 9

Best Known (109−59, 109, s)-Nets in Base 9

(109−59, 109, 104)-Net over F₉ — Constructive and digital

Digital (50, 109, 104)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (8, 37, 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, 72, 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]

(109−59, 109, 182)-Net over F₉ — Digital

Digital (50, 109, 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]

(109−59, 109, 5203)-Net in Base 9 — Upper bound on s

There is no (50, 109, 5204)-net in base 9, because

1 times m-reduction [i] would yield (50, 108, 5204)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 11 465500 289863 399149 800885 557015 691828 225921 733642 821695 730981 656655 462112 457982 136072 736445 260277 554977 > 9¹⁰⁸ [i]