MinT - Best Known (65−20, 65, s)-Nets in Base 8

Best Known (65−20, 65, s)-Nets in Base 8

(65−20, 65, 368)-Net over F₈ — Constructive and digital

Digital (45, 65, 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 (34, 54, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 27, 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]

(65−20, 65, 528)-Net in Base 8 — Constructive

(45, 65, 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. (34, 54, 514)-net in base 8, using
  - trace code for nets [i] based on (7, 27, 257)-net in base 64, using
    - 1 times m-reduction [i] based on (7, 28, 257)-net in base 64, using
      - base change [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]

(65−20, 65, 1401)-Net over F₈ — Digital

Digital (45, 65, 1401)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(65−20, 65, 479686)-Net in Base 8 — Upper bound on s

There is no (45, 65, 479687)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 50216 972972 355565 319857 959409 071056 657746 890272 866351 191432 > 8⁶⁵ [i]