MinT - Best Known (69−21, 69, s)-Nets in Base 8

Best Known (69−21, 69, s)-Nets in Base 8

(69−21, 69, 378)-Net over F₈ — Constructive and digital

Digital (48, 69, 378)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (3, 13, 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 (35, 56, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 28, 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]

(69−21, 69, 538)-Net in Base 8 — Constructive

(48, 69, 538)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. digital (3, 13, 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. (35, 56, 514)-net in base 8, using
  - base change [i] based on digital (21, 42, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 21, 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]

(69−21, 69, 1558)-Net over F₈ — Digital

Digital (48, 69, 1558)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(69−21, 69, 895132)-Net in Base 8 — Upper bound on s

There is no (48, 69, 895133)-net in base 8, because

1 times m-reduction [i] would yield (48, 68, 895133)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 25 711168 367608 877422 704963 069031 573273 323702 243817 618632 641766 > 8⁶⁸ [i]