MinT - Best Known (90−49, 90, s)-Nets in Base 9

Best Known (90−49, 90, s)-Nets in Base 9

(90−49, 90, 94)-Net over F₉ — Constructive and digital

Digital (41, 90, 94)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (4, 28, 30)-net over F₉, using
  - net from sequence [i] based on digital (4, 29)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 4 and N(F) ≥ 30, using
      - a function field by SÃ©mirat [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]

(90−49, 90, 96)-Net in Base 9 — Constructive

(41, 90, 96)-net in base 9, using

base change [i] based on digital (11, 60, 96)-net over F₂₇, using
- net from sequence [i] based on digital (11, 95)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 11 and N(F) ≥ 96, using
    - an explicitly constructive algebraic function field [i]

(90−49, 90, 140)-Net over F₉ — Digital

Digital (41, 90, 140)-net over F₉, using

t-expansion [i] based on 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]

(90−49, 90, 4221)-Net in Base 9 — Upper bound on s

There is no (41, 90, 4222)-net in base 9, because

1 times m-reduction [i] would yield (41, 89, 4222)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 8 503362 047650 416814 675321 319282 811795 367915 630266 231960 310113 129159 338412 025697 330049 > 9⁸⁹ [i]