MinT - Best Known (89−49, 89, s)-Nets in Base 9

Best Known (89−49, 89, s)-Nets in Base 9

(89−49, 89, 92)-Net over F₉ — Constructive and digital

Digital (40, 89, 92)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (3, 27, 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, 62, 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]

(89−49, 89, 94)-Net in Base 9 — Constructive

(40, 89, 94)-net in base 9, using

1 times m-reduction [i] based on (40, 90, 94)-net in base 9, using
- base change [i] based on digital (10, 60, 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]

(89−49, 89, 140)-Net over F₉ — Digital

Digital (40, 89, 140)-net over F₉, using

t-expansion [i] based on digital (39, 89, 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]

(89−49, 89, 3850)-Net in Base 9 — Upper bound on s

There is no (40, 89, 3851)-net in base 9, because

1 times m-reduction [i] would yield (40, 88, 3851)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 942641 503213 965804 956225 834541 767594 070181 023012 242894 366034 474102 154252 998285 922881 > 9⁸⁸ [i]