MinT - Best Known (50, 50+69, s)-Nets in Base 9

Best Known (50, 50+69, s)-Nets in Base 9

(50, 50+69, 92)-Net over F₉ — Constructive and digital

Digital (50, 119, 92)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (3, 37, 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]
2. digital (13, 82, 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]

(50, 50+69, 94)-Net in Base 9 — Constructive

(50, 119, 94)-net in base 9, using

1 times m-reduction [i] based on (50, 120, 94)-net in base 9, using
- base change [i] based on digital (10, 80, 94)-net over F₂₇, using
  - net from sequence [i] based on digital (10, 93)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 10 and N(F) ≥ 94, using
      - a function field by SÃ©mirat [i]

(50, 50+69, 182)-Net over F₉ — Digital

Digital (50, 119, 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]

(50, 50+69, 3447)-Net in Base 9 — Upper bound on s

There is no (50, 119, 3448)-net in base 9, because

1 times m-reduction [i] would yield (50, 118, 3448)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 39867 725841 661324 497281 607966 818116 258937 079723 786706 261281 948885 703328 282168 436353 618294 195656 071400 136662 734465 > 9¹¹⁸ [i]