MinT - Best Known (39, 39+18, s)-Nets in Base 8

Best Known (39, 39+18, s)-Nets in Base 8

(39, 39+18, 354)-Net over F₈ — Constructive and digital

Digital (39, 57, 354)-net over F₈, using

7 times m-reduction [i] based on digital (39, 64, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 32, 177)-net over F₆₄, using
  - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
      - a function field by GarcÃa and Quoos [i]

(39, 39+18, 523)-Net in Base 8 — Constructive

(39, 57, 523)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. digital (0, 9, 9)-net over F₈, using
  - net from sequence [i] based on digital (0, 8)-sequence over F₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 0 and N(F) ≥ 9, using
      - the rational function field F₈(x) [i]
    - Niederreiter sequence [i]
  - a shift-net [i]
2. (30, 48, 514)-net in base 8, using
  - base change [i] based on digital (18, 36, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 18, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]

(39, 39+18, 1102)-Net over F₈ — Digital

Digital (39, 57, 1102)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(39, 39+18, 310610)-Net in Base 8 — Upper bound on s

There is no (39, 57, 310611)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 2993 210958 860632 412527 544219 697724 633505 189977 764212 > 8⁵⁷ [i]