MinT - Best Known (58−18, 58, s)-Nets in Base 9

Best Known (58−18, 58, s)-Nets in Base 9

(58−18, 58, 400)-Net over F₉ — Constructive and digital

Digital (40, 58, 400)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (11, 20, 200)-net over F₉, using
  - trace code for nets [i] based on digital (1, 10, 100)-net over F₈₁, using
    - net from sequence [i] based on digital (1, 99)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 1 and N(F) ≥ 100, using
        a function field by SÃ©mirat [i]
2. digital (20, 38, 200)-net over F₉, using
  - trace code for nets [i] based on digital (1, 19, 100)-net over F₈₁, using
    - net from sequence [i] based on digital (1, 99)-sequence over F₈₁ (see above)

(58−18, 58, 1625)-Net over F₉ — Digital

Digital (40, 58, 1625)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(58−18, 58, 731511)-Net in Base 9 — Upper bound on s

There is no (40, 58, 731512)-net in base 9, because

the generalized Rao bound for nets shows that 9^mÂ â‰¥ 22 185403 284783 665890 733083 608681 340477 742980 780359 280065 > 9⁵⁸ [i]