MinT - Best Known (42, 61, s)-Nets in Base 8

Best Known (42, 61, s)-Nets in Base 8

(42, 61, 363)-Net over F₈ — Constructive and digital

Digital (42, 61, 363)-net over F₈, 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. digital (33, 52, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 26, 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]

(42, 61, 523)-Net in Base 8 — Constructive

(42, 61, 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. (33, 52, 514)-net in base 8, using
  - base change [i] based on digital (20, 39, 514)-net over F₁₆, using
    - 1 times m-reduction [i] based on digital (20, 40, 514)-net over F₁₆, using
      - trace code for nets [i] based on digital (0, 20, 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]

(42, 61, 1249)-Net over F₈ — Digital

Digital (42, 61, 1249)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(42, 61, 621226)-Net in Base 8 — Upper bound on s

There is no (42, 61, 621227)-net in base 8, because

1 times m-reduction [i] would yield (42, 60, 621227)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1 532517 668456 449240 973826 696853 986771 537504 272133 796566 > 8⁶⁰ [i]