MinT - Best Known (136−77, 136, s)-Nets in Base 9

Best Known (136−77, 136, s)-Nets in Base 9

(136−77, 136, 104)-Net over F₉ — Constructive and digital

Digital (59, 136, 104)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (8, 46, 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, 90, 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]

(136−77, 136, 182)-Net over F₉ — Digital

Digital (59, 136, 182)-net over F₉, using

t-expansion [i] based on digital (50, 136, 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]

(136−77, 136, 4587)-Net in Base 9 — Upper bound on s

There is no (59, 136, 4588)-net in base 9, because

1 times m-reduction [i] would yield (59, 135, 4588)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 666 630660 643936 221587 509607 009704 523102 963227 212783 811652 416171 912064 230458 634472 468936 714475 556084 014639 143275 178277 970542 528705 > 9¹³⁵ [i]