MinT - Best Known (43−13, 43, s)-Nets in Base 8

Best Known (43−13, 43, s)-Nets in Base 8

(43−13, 43, 354)-Net over F₈ — Constructive and digital

Digital (30, 43, 354)-net over F₈, using

3 times m-reduction [i] based on digital (30, 46, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 23, 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]

(43−13, 43, 528)-Net in Base 8 — Constructive

(30, 43, 528)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. digital (1, 7, 14)-net over F₈, using
  - net from sequence [i] based on digital (1, 13)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 1 and N(F) ≥ 14, using
      - F₂ from the tower of function fields over F₈ by van der Geer and van der Vlugt [i]
2. (23, 36, 514)-net in base 8, using
  - base change [i] based on digital (14, 27, 514)-net over F₁₆, using
    - 1 times m-reduction [i] based on digital (14, 28, 514)-net over F₁₆, using
      - trace code for nets [i] based on digital (0, 14, 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]

(43−13, 43, 1307)-Net over F₈ — Digital

Digital (30, 43, 1307)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(43−13, 43, 896917)-Net in Base 8 — Upper bound on s

There is no (30, 43, 896918)-net in base 8, because

1 times m-reduction [i] would yield (30, 42, 896918)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 85 071077 973501 428857 865958 520978 488968 > 8⁴² [i]