MinT - Best Known (122−36, 122, s)-Nets in Base 9

Best Known (122−36, 122, s)-Nets in Base 9

(122−36, 122, 750)-Net over F₉ — Constructive and digital

Digital (86, 122, 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 (68, 104, 740)-net over F₉, using
  - trace code for nets [i] based on digital (16, 52, 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]

(122−36, 122, 3702)-Net over F₉ — Digital

Digital (86, 122, 3702)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(122−36, 122, 2771290)-Net in Base 9 — Upper bound on s

There is no (86, 122, 2771291)-net in base 9, because

the generalized Rao bound for nets shows that 9^mÂ â‰¥ 261 569699 688444 086427 785131 401125 981383 663450 139533 804031 217971 879788 095908 738268 855847 246817 106060 900769 941374 244273 > 9¹²² [i]