MinT - Best Known (136−79, 136, s)-Nets in Base 9

Best Known (136−79, 136, s)-Nets in Base 9

(136−79, 136, 96)-Net over F₉ — Constructive and digital

Digital (57, 136, 96)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (5, 44, 32)-net over F₉, using
  - net from sequence [i] based on digital (5, 31)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 5 and N(F) ≥ 32, using
      - a function field by GarcÃa and Quoos [i]
2. digital (13, 92, 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]

(136−79, 136, 182)-Net over F₉ — Digital

Digital (57, 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]

(136−79, 136, 3844)-Net in Base 9 — Upper bound on s

There is no (57, 136, 3845)-net in base 9, because

1 times m-reduction [i] would yield (57, 135, 3845)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 670 715795 379844 918388 198018 002610 887416 370893 845625 273920 442576 517118 347463 810053 971814 812961 366781 194595 326376 248824 867466 992345 > 9¹³⁵ [i]