MinT - Best Known (76, 76+27, s)-Nets in Base 8

Best Known (76, 76+27, s)-Nets in Base 8

(76, 76+27, 562)-Net over F₈ — Constructive and digital

Digital (76, 103, 562)-net over F₈, using

1 times m-reduction [i] based on digital (76, 104, 562)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (20, 34, 208)-net over F₈, using
    - trace code for nets [i] based on digital (3, 17, 104)-net over F₆₄, using
      - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]
  2. digital (42, 70, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 35, 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]

(76, 76+27, 674)-Net in Base 8 — Constructive

(76, 103, 674)-net in base 8, using

8³ times duplication [i] based on (73, 100, 674)-net in base 8, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 28, 160)-net over F₈, using
    - trace code for nets [i] based on digital (1, 14, 80)-net over F₆₄, using
      - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
        a function field by SÃ©mirat [i]
  2. (45, 72, 514)-net in base 8, using
    - base change [i] based on digital (27, 54, 514)-net over F₁₆, using
      - trace code for nets [i] based on digital (0, 27, 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]

(76, 76+27, 5712)-Net over F₈ — Digital

Digital (76, 103, 5712)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(76, 76+27, large)-Net in Base 8 — Upper bound on s

There is no (76, 103, large)-net in base 8, because

25 times m-reduction [i] would yield (76, 78, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]