MinT - Best Known (31, 31+13, s)-Nets in Base 8

Best Known (31, 31+13, s)-Nets in Base 8

(31, 31+13, 354)-Net over F₈ — Constructive and digital

Digital (31, 44, 354)-net over F₈, using

4 times m-reduction [i] based on digital (31, 48, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 24, 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]

(31, 31+13, 538)-Net in Base 8 — Constructive

(31, 44, 538)-net in base 8, using

base change [i] based on digital (20, 33, 538)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (1, 7, 24)-net over F₁₆, using
    - net from sequence [i] based on digital (1, 23)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 1 and N(F) ≥ 24, using
        a function field by SÃ©mirat [i]
  2. digital (13, 26, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 13, 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]

(31, 31+13, 1553)-Net over F₈ — Digital

Digital (31, 44, 1553)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(31, 31+13, 1268433)-Net in Base 8 — Upper bound on s

There is no (31, 44, 1268434)-net in base 8, because

1 times m-reduction [i] would yield (31, 43, 1268434)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 680 565435 109296 055602 926282 893398 927160 > 8⁴³ [i]