MinT - Best Known (87, 87+37, s)-Nets in Base 9

Best Known (87, 87+37, s)-Nets in Base 9

(87, 87+37, 750)-Net over F₉ — Constructive and digital

Digital (87, 124, 750)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (0, 18, 10)-net over F₉, using
  - net from sequence [i] based on digital (0, 9)-sequence over F₉, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 0 and N(F) ≥ 10, using
      - the rational function field F₉(x) [i]
    - Niederreiter sequence [i]
2. digital (69, 106, 740)-net over F₉, using
  - trace code for nets [i] based on digital (16, 53, 370)-net over F₈₁, using
    - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
        a function field by GarcÃa and Quoos [i]

(87, 87+37, 3473)-Net over F₉ — Digital

Digital (87, 124, 3473)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(87, 87+37, 3131091)-Net in Base 9 — Upper bound on s

There is no (87, 124, 3131092)-net in base 9, because

1 times m-reduction [i] would yield (87, 123, 3131092)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 2354 130308 580992 723184 480396 311170 812344 088240 862831 320009 801170 463168 720072 740658 328499 691191 064443 492589 855904 518081 > 9¹²³ [i]