MinT - Best Known (90−51, 90, s)-Nets in Base 9

Best Known (90−51, 90, s)-Nets in Base 9

(90−51, 90, 81)-Net over F₉ — Constructive and digital

Digital (39, 90, 81)-net over F₉, using

t-expansion [i] based on digital (32, 90, 81)-net over F₉, using
- net from sequence [i] based on digital (32, 80)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 32 and N(F) ≥ 81, using
    - F₄ from the tower of function fields by Bezerra and GarcÃa over F₉ [i]

(90−51, 90, 88)-Net in Base 9 — Constructive

(39, 90, 88)-net in base 9, using

base change [i] based on digital (9, 60, 88)-net over F₂₇, using
- net from sequence [i] based on digital (9, 87)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 9 and N(F) ≥ 88, using
    - a function field by SÃ©mirat [i]

(90−51, 90, 140)-Net over F₉ — Digital

Digital (39, 90, 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]

(90−51, 90, 3159)-Net in Base 9 — Upper bound on s

There is no (39, 90, 3160)-net in base 9, because

1 times m-reduction [i] would yield (39, 89, 3160)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 8 522193 357281 854451 606299 520463 078376 504379 852967 539863 395348 556096 546964 787877 013697 > 9⁸⁹ [i]