MinT - Best Known (64−19, 64, s)-Nets in Base 8

Best Known (64−19, 64, s)-Nets in Base 8

(64−19, 64, 378)-Net over F₈ — Constructive and digital

Digital (45, 64, 378)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (3, 12, 24)-net over F₈, using
  - net from sequence [i] based on digital (3, 23)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 3 and N(F) ≥ 24, using
      - the Klein quartic over F₈ [i]
2. digital (33, 52, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 26, 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]

(64−19, 64, 538)-Net in Base 8 — Constructive

(45, 64, 538)-net in base 8, using

base change [i] based on digital (29, 48, 538)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (1, 10, 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 (19, 38, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 19, 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]

(64−19, 64, 1763)-Net over F₈ — Digital

Digital (45, 64, 1763)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(64−19, 64, 1242457)-Net in Base 8 — Upper bound on s

There is no (45, 64, 1242458)-net in base 8, because

1 times m-reduction [i] would yield (45, 63, 1242458)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 784 641738 744953 506867 515131 623970 635210 091710 195636 975705 > 8⁶³ [i]