MinT - Best Known (109−28, 109, s)-Nets in Base 8

Best Known (109−28, 109, s)-Nets in Base 8

(109−28, 109, 610)-Net over F₈ — Constructive and digital

Digital (81, 109, 610)-net over F₈, using

1 times m-reduction [i] based on digital (81, 110, 610)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (24, 38, 256)-net over F₈, using
    - trace code for nets [i] based on digital (5, 19, 128)-net over F₆₄, using
      - net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
        a function field by SÃ©mirat [i]
  2. digital (43, 72, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 36, 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]

(109−28, 109, 708)-Net in Base 8 — Constructive

(81, 109, 708)-net in base 8, using

1 times m-reduction [i] based on (81, 110, 708)-net in base 8, using
- (u,Â u+v)-construction [i] based on
  1. (24, 38, 514)-net in base 8, using
    - trace code for nets [i] based on (5, 19, 257)-net in base 64, using
      - 1 times m-reduction [i] based on (5, 20, 257)-net in base 64, using
        base change [i] based on digital (0, 15, 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]
  2. digital (43, 72, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 36, 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]

(109−28, 109, 6918)-Net over F₈ — Digital

Digital (81, 109, 6918)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(109−28, 109, large)-Net in Base 8 — Upper bound on s

There is no (81, 109, large)-net in base 8, because

26 times m-reduction [i] would yield (81, 83, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]