MinT - Best Known (40, 40+19, s)-Nets in Base 9

Best Known (40, 40+19, s)-Nets in Base 9

(40, 40+19, 364)-Net over F₉ — Constructive and digital

Digital (40, 59, 364)-net over F₉, using

9¹ times duplication [i] based on digital (39, 58, 364)-net over F₉, using
- (u,Â u+v)-construction [i] based on
  1. digital (9, 18, 164)-net over F₉, using
    - trace code for nets [i] based on digital (0, 9, 82)-net over F₈₁, using
      - net from sequence [i] based on digital (0, 81)-sequence over F₈₁, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 0 and N(F) ≥ 82, using
        the rational function field F₈₁(x) [i]
        Niederreiter sequence [i]
  2. digital (21, 40, 200)-net over F₉, using
    - trace code for nets [i] based on digital (1, 20, 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]

(40, 40+19, 1276)-Net over F₉ — Digital

Digital (40, 59, 1276)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(40, 40+19, 731511)-Net in Base 9 — Upper bound on s

There is no (40, 59, 731512)-net in base 9, because

1 times m-reduction [i] would yield (40, 58, 731512)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 22 185403 284783 665890 733083 608681 340477 742980 780359 280065 > 9⁵⁸ [i]