MinT - Best Known (66−21, 66, s)-Nets in Base 8

Best Known (66−21, 66, s)-Nets in Base 8

(66−21, 66, 363)-Net over F₈ — Constructive and digital

Digital (45, 66, 363)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 10, 9)-net over F₈, using
  - net from sequence [i] based on digital (0, 8)-sequence over F₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 0 and N(F) ≥ 9, using
      - the rational function field F₈(x) [i]
    - Niederreiter sequence [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]

(66−21, 66, 523)-Net in Base 8 — Constructive

(45, 66, 523)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. digital (0, 10, 9)-net over F₈, using
  - net from sequence [i] based on digital (0, 8)-sequence over F₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 0 and N(F) ≥ 9, using
      - the rational function field F₈(x) [i]
    - Niederreiter sequence [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]

(66−21, 66, 1143)-Net over F₈ — Digital

Digital (45, 66, 1143)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(66−21, 66, 479686)-Net in Base 8 — Upper bound on s

There is no (45, 66, 479687)-net in base 8, because

1 times m-reduction [i] would yield (45, 65, 479687)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 50216 972972 355565 319857 959409 071056 657746 890272 866351 191432 > 8⁶⁵ [i]