MinT - Best Known (41, 41+51, s)-Nets in Base 9

Best Known (41, 41+51, s)-Nets in Base 9

(41, 41+51, 92)-Net over F₉ — Constructive and digital

Digital (41, 92, 92)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (3, 28, 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, 64, 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]

(41, 41+51, 94)-Net in Base 9 — Constructive

(41, 92, 94)-net in base 9, using

1 times m-reduction [i] based on (41, 93, 94)-net in base 9, using
- base change [i] based on digital (10, 62, 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]

(41, 41+51, 140)-Net over F₉ — Digital

Digital (41, 92, 140)-net over F₉, using

t-expansion [i] based on digital (39, 92, 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]

(41, 41+51, 3769)-Net in Base 9 — Upper bound on s

There is no (41, 92, 3770)-net in base 9, because

1 times m-reduction [i] would yield (41, 91, 3770)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 689 537766 995523 835654 256661 848506 935774 947904 522268 444243 269903 568210 641221 298695 013969 > 9⁹¹ [i]