MinT - Best Known (128−38, 128, s)-Nets in Base 9

Best Known (128−38, 128, s)-Nets in Base 9

(128−38, 128, 756)-Net over F₉ — Constructive and digital

Digital (90, 128, 756)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (1, 20, 16)-net over F₉, using
  - net from sequence [i] based on digital (1, 15)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 1 and N(F) ≥ 16, using
      - a function field by GarcÃa and Quoos [i]
2. digital (70, 108, 740)-net over F₉, using
  - trace code for nets [i] based on digital (16, 54, 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]

(128−38, 128, 3683)-Net over F₉ — Digital

Digital (90, 128, 3683)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(128−38, 128, 2658916)-Net in Base 9 — Upper bound on s

There is no (90, 128, 2658917)-net in base 9, because

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]