MinT - Best Known (129−39, 129, s)-Nets in Base 9

Best Known (129−39, 129, s)-Nets in Base 9

(129−39, 129, 750)-Net over F₉ — Constructive and digital

Digital (90, 129, 750)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (0, 19, 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 (71, 110, 740)-net over F₉, using
  - trace code for nets [i] based on digital (16, 55, 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]

(129−39, 129, 3278)-Net over F₉ — Digital

Digital (90, 129, 3278)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(129−39, 129, 2658916)-Net in Base 9 — Upper bound on s

There is no (90, 129, 2658917)-net in base 9, because

1 times m-reduction [i] would yield (90, 128, 2658917)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 139 008470 922597 173529 734528 714693 973067 459147 074137 475510 771348 350974 993027 418228 622248 947165 341294 032074 352465 640438 771513 > 9¹²⁸ [i]