MinT - Best Known (12, 12+17, s)-Nets in Base 9

Best Known (12, 12+17, s)-Nets in Base 9

(12, 12+17, 44)-Net over F₉ — Constructive and digital

Digital (12, 29, 44)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (1, 9, 16)-net over F₉, using
  - net from sequence [i] based on digital (1, 15)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 1 and N(F) ≥ 16, using
      - a function field by GarcÃa and Quoos [i]
2. digital (3, 20, 28)-net over F₉, using
  - net from sequence [i] based on digital (3, 27)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 3 and N(F) ≥ 28, using
      - the Hermitian function field over F₉ [i]

(12, 12+17, 48)-Net in Base 9 — Constructive

(12, 29, 48)-net in base 9, using

1 times m-reduction [i] based on (12, 30, 48)-net in base 9, using
- base change [i] based on digital (2, 20, 48)-net over F₂₇, using
  - net from sequence [i] based on digital (2, 47)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 2 and N(F) ≥ 48, using
      - a function field by GarcÃa and Quoos [i]

(12, 12+17, 56)-Net over F₉ — Digital

Digital (12, 29, 56)-net over F₉, using

net from sequence [i] based on digital (12, 55)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 12 and N(F) ≥ 56, using
  - a curve from the manYPoints database [i]

(12, 12+17, 1024)-Net in Base 9 — Upper bound on s

There is no (12, 29, 1025)-net in base 9, because

1 times m-reduction [i] would yield (12, 28, 1025)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 525 035535 881708 014315 231041 > 9²⁸ [i]