MinT - Best Known (67−21, 67, s)-Nets in Base 8

Best Known (67−21, 67, s)-Nets in Base 8

(67−21, 67, 368)-Net over F₈ — Constructive and digital

Digital (46, 67, 368)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 11, 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. 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]

(67−21, 67, 528)-Net in Base 8 — Constructive

(46, 67, 528)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. digital (1, 11, 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. (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]

(67−21, 67, 1268)-Net over F₈ — Digital

Digital (46, 67, 1268)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(67−21, 67, 590565)-Net in Base 8 — Upper bound on s

There is no (46, 67, 590566)-net in base 8, because

1 times m-reduction [i] would yield (46, 66, 590566)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 401740 269704 170815 321291 376586 799031 867046 567774 538493 056872 > 8⁶⁶ [i]