MinT - Best Known (12, 12+15, s)-Nets in Base 9

Best Known (12, 12+15, s)-Nets in Base 9

(12, 12+15, 48)-Net over F₉ — Constructive and digital

Digital (12, 27, 48)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (2, 9, 20)-net over F₉, using
  - net from sequence [i] based on digital (2, 19)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 2 and N(F) ≥ 20, using
      - a function field by GarcÃa and Quoos [i]
2. digital (3, 18, 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+15, 52)-Net in Base 9 — Constructive

(12, 27, 52)-net in base 9, using

base change [i] based on digital (3, 18, 52)-net over F₂₇, using
- net from sequence [i] based on digital (3, 51)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 3 and N(F) ≥ 52, using
    - a function field by SÃ©mirat [i]

(12, 12+15, 56)-Net over F₉ — Digital

Digital (12, 27, 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+15, 1475)-Net in Base 9 — Upper bound on s

There is no (12, 27, 1476)-net in base 9, because

1 times m-reduction [i] would yield (12, 26, 1476)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 6 467966 138585 758626 469729 > 9²⁶ [i]