MinT - Best Known (37, 37+17, s)-Nets in Base 8

Best Known (37, 37+17, s)-Nets in Base 8

(37, 37+17, 354)-Net over F₈ — Constructive and digital

Digital (37, 54, 354)-net over F₈, using

6 times m-reduction [i] based on digital (37, 60, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 30, 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]

(37, 37+17, 523)-Net in Base 8 — Constructive

(37, 54, 523)-net in base 8, 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. (29, 46, 514)-net in base 8, using
  - trace code for nets [i] based on (6, 23, 257)-net in base 64, using
    - 1 times m-reduction [i] based on (6, 24, 257)-net in base 64, using
      - base change [i] based on digital (0, 18, 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]

(37, 37+17, 1093)-Net over F₈ — Digital

Digital (37, 54, 1093)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(37, 37+17, 517081)-Net in Base 8 — Upper bound on s

There is no (37, 54, 517082)-net in base 8, because

1 times m-reduction [i] would yield (37, 53, 517082)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 730752 570932 355983 302424 141068 221049 824175 864975 > 8⁵³ [i]