MinT - Best Known (49−15, 49, s)-Nets in Base 8

Best Known (49−15, 49, s)-Nets in Base 8

(49−15, 49, 354)-Net over F₈ — Constructive and digital

Digital (34, 49, 354)-net over F₈, using

5 times m-reduction [i] based on digital (34, 54, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 27, 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]

(49−15, 49, 531)-Net in Base 8 — Constructive

(34, 49, 531)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. digital (2, 9, 17)-net over F₈, using
  - net from sequence [i] based on digital (2, 16)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 2 and N(F) ≥ 17, using
      - a function field by SÃ©mirat [i]
2. (25, 40, 514)-net in base 8, using
  - base change [i] based on digital (15, 30, 514)-net over F₁₆, using
    - trace code for nets [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]

(49−15, 49, 1258)-Net over F₈ — Digital

Digital (34, 49, 1258)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(49−15, 49, 752376)-Net in Base 8 — Upper bound on s

There is no (34, 49, 752377)-net in base 8, because

1 times m-reduction [i] would yield (34, 48, 752377)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 22 300801 989165 330862 600825 500293 350087 750408 > 8⁴⁸ [i]