MinT - Best Known (40, 57, s)-Nets in Base 9

Best Known (40, 57, s)-Nets in Base 9

(40, 57, 432)-Net over F₉ — Constructive and digital

Digital (40, 57, 432)-net over F₉, using

9¹ times duplication [i] based on digital (39, 56, 432)-net over F₉, using
- (u,Â u+v)-construction [i] based on
  1. digital (10, 18, 200)-net over F₉, using
    - trace code for nets [i] based on digital (1, 9, 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 (21, 38, 232)-net over F₉, using
    - trace code for nets [i] based on digital (2, 19, 116)-net over F₈₁, using
      - net from sequence [i] based on digital (2, 115)-sequence over F₈₁, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 116, using
        a function field by SÃ©mirat [i]

(40, 57, 2141)-Net over F₉ — Digital

Digital (40, 57, 2141)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(40, 57, 2250592)-Net in Base 9 — Upper bound on s

There is no (40, 57, 2250593)-net in base 9, because

1 times m-reduction [i] would yield (40, 56, 2250593)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 273893 691492 235418 615213 655921 728001 104040 179198 467905 > 9⁵⁶ [i]