MinT - Best Known (70−23, 70, s)-Nets in Base 9

Best Known (70−23, 70, s)-Nets in Base 9

(70−23, 70, 364)-Net over F₉ — Constructive and digital

Digital (47, 70, 364)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (11, 22, 164)-net over F₉, using
  - trace code for nets [i] based on digital (0, 11, 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 (25, 48, 200)-net over F₉, using
  - trace code for nets [i] based on digital (1, 24, 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]

(70−23, 70, 1241)-Net over F₉ — Digital

Digital (47, 70, 1241)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(70−23, 70, 593775)-Net in Base 9 — Upper bound on s

There is no (47, 70, 593776)-net in base 9, because

1 times m-reduction [i] would yield (47, 69, 593776)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 696200 716367 742946 018934 317889 057916 895319 248181 543551 970802 462849 > 9⁶⁹ [i]