MinT - Best Known (56−17, 56, s)-Nets in Base 8

Best Known (56−17, 56, s)-Nets in Base 8

(56−17, 56, 363)-Net over F₈ — Constructive and digital

Digital (39, 56, 363)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 8, 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]
2. digital (31, 48, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 24, 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]

(56−17, 56, 531)-Net in Base 8 — Constructive

(39, 56, 531)-net in base 8, using

base change [i] based on digital (25, 42, 531)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (0, 8, 17)-net over F₁₆, using
    - net from sequence [i] based on digital (0, 16)-sequence over F₁₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 0 and N(F) ≥ 17, using
        the rational function field F₁₆(x) [i]
      - Niederreiter sequence [i]
  2. digital (17, 34, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 17, 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]

(56−17, 56, 1415)-Net over F₈ — Digital

Digital (39, 56, 1415)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(56−17, 56, 869627)-Net in Base 8 — Upper bound on s

There is no (39, 56, 869628)-net in base 8, because

1 times m-reduction [i] would yield (39, 55, 869628)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 46 768179 238168 640081 120064 851782 708570 955494 048783 > 8⁵⁵ [i]