MinT - Best Known (82−43, 82, s)-Nets in Base 9

Best Known (82−43, 82, s)-Nets in Base 9

(82−43, 82, 96)-Net over F₉ — Constructive and digital

Digital (39, 82, 96)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (5, 26, 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, 56, 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]

(82−43, 82, 140)-Net over F₉ — Digital

Digital (39, 82, 140)-net over F₉, using

net from sequence [i] based on digital (39, 139)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 39 and N(F) ≥ 140, using
  - a curve from the manYPoints database [i]

(82−43, 82, 5187)-Net in Base 9 — Upper bound on s

There is no (39, 82, 5188)-net in base 9, because

1 times m-reduction [i] would yield (39, 81, 5188)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 196650 935436 024899 893713 778838 613970 519808 212510 654659 635276 505985 151860 697505 > 9⁸¹ [i]