MinT - Best Known (113−29, 113, s)-Nets in Base 8

Best Known (113−29, 113, s)-Nets in Base 8

(113−29, 113, 610)-Net over F₈ — Constructive and digital

Digital (84, 113, 610)-net over F₈, using

1 times m-reduction [i] based on digital (84, 114, 610)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (25, 40, 256)-net over F₈, using
    - trace code for nets [i] based on digital (5, 20, 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 (44, 74, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 37, 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]

(113−29, 113, 772)-Net in Base 8 — Constructive

(84, 113, 772)-net in base 8, using

8¹ times duplication [i] based on (83, 112, 772)-net in base 8, using
- (u,Â u+v)-construction [i] based on
  1. (20, 34, 258)-net in base 8, using
    - trace code for nets [i] based on (3, 17, 129)-net in base 64, using
      - 4 times m-reduction [i] based on (3, 21, 129)-net in base 64, using
        base change [i] based on digital (0, 18, 129)-net over F₁₂₈, using
        net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
        the rational function field F₁₂₈(x) [i]
        Niederreiter sequence [i]
  2. (49, 78, 514)-net in base 8, using
    - trace code for nets [i] based on (10, 39, 257)-net in base 64, using
      - 1 times m-reduction [i] based on (10, 40, 257)-net in base 64, using
        base change [i] based on digital (0, 30, 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]

(113−29, 113, 7135)-Net over F₈ — Digital

Digital (84, 113, 7135)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(113−29, 113, large)-Net in Base 8 — Upper bound on s

There is no (84, 113, large)-net in base 8, because

27 times m-reduction [i] would yield (84, 86, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]